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Joshua Wiscons

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California State University, Sacramento

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Conference in honour of 60th birthdays of Alexandre Borovik and Ali Nesin

BN-Pair 19-22 October 2016, Istanbul

Conference in honour of 60th birthdays of Alexandre Borovik and Ali Nesin 19-22 October 2016, Istanbul

Conference Organizers: Tuna Altınel (Lyon), Ayşe Berkman (İstanbul), Özlem Beyarslan (İstanbul),

October, 2016

Organizers: Tuna Altınel (Lyon), Ayşe Berkman (İstanbul), Özlem Beyarslan (İstanbul),

Joint work with Adrien Deloro

Based upon work supported by NSF grant No. OISE-1064446. Joshua Wiscons

Geometries and small groups of fMr

Act I Background on groups of finite Morley rank

Joshua Wiscons

Geometries and small groups of fMr

Groups of finite Morley rank (fMr)

Groups of fMr

Joshua Wiscons

Geometries and small groups of fMr

Groups of finite Morley rank (fMr)

Groups of fMr

Affine algebraic groups

Joshua Wiscons

Geometries and small groups of fMr

Groups of finite Morley rank (fMr)

Groups of fMr

Affine algebraic groups PGLn (K) GLn (K)

Joshua Wiscons

Geometries and small groups of fMr

Groups of finite Morley rank (fMr)

Groups of fMr Zp∞ GLn (K1 ) × GLn (K2 )

Affine algebraic groups PGLn (K) GLn (K)

Joshua Wiscons

Geometries and small groups of fMr

Groups of finite Morley rank (fMr)

Groups of fMr Simple groups of fMr

?

Zp∞

GLn (K1 ) × GLn (K2 )

Affine algebraic groups PGLn (K) GLn (K)

Joshua Wiscons

Geometries and small groups of fMr

Groups of finite Morley rank (fMr)

Groups of fMr Simple groups of fMr Zp∞

? GLn (K1 ) × GLn (K2 )

Affine algebraic groups PGLn (K) GLn (K)

Algebraicity Conjecture: the gap, Joshua Wiscons

, does not exist. Geometries and small groups of fMr

Groups of finite Morley rank (fMr)

Groups of fMr

Affine algebraic groups

Joshua Wiscons

Geometries and small groups of fMr

The world

Joshua Wiscons

Geometries and small groups of fMr

The world

fMr

Joshua Wiscons

Geometries and small groups of fMr

The world

NTP2 NIP fMr Stable

ω-stable

Simple

Joshua Wiscons

Geometries and small groups of fMr

The world

NTP2 NIP fMr

RCF

Stable Pseudo-Finite Fields ω-stable

Fn

Simple

Joshua Wiscons

Geometries and small groups of fMr

Algebraicity Conjecture Algebraicity Conjecture (Cherlin-Zilber) An infinite simple group of finite Morley rank is isomorphic to an (affine) algebraic group over an algebraically closed field.

Joshua Wiscons

Geometries and small groups of fMr

Algebraicity Conjecture Algebraicity Conjecture (Cherlin-Zilber) An infinite simple group of finite Morley rank is isomorphic to an (affine) algebraic group over an algebraically closed field. Analysis of groups of fMr breaks into 4 types.

Joshua Wiscons

Geometries and small groups of fMr

Algebraicity Conjecture Algebraicity Conjecture (Cherlin-Zilber) An infinite simple group of finite Morley rank is isomorphic to an (affine) algebraic group over an algebraically closed field. Analysis of groups of fMr breaks into 4 types.

does NOT contain

L

Z2

contains

L

Z2

i<ω

i<ω

Joshua Wiscons

Geometries and small groups of fMr

Algebraicity Conjecture Algebraicity Conjecture (Cherlin-Zilber) An infinite simple group of finite Morley rank is isomorphic to an (affine) algebraic group over an algebraically closed field. Analysis of groups of fMr breaks into 4 types. does NOT contain Z2∞ does NOT contain

L

Z2

contains

L

Z2

i<ω

i<ω

Joshua Wiscons

contains Z2∞

Geometries and small groups of fMr

Algebraicity Conjecture Algebraicity Conjecture (Cherlin-Zilber) An infinite simple group of finite Morley rank is isomorphic to an (affine) algebraic group over an algebraically closed field. Analysis of groups of fMr breaks into 4 types. does NOT contain Z2∞ does NOT contain

L

Z2

contains

L

Z2

i<ω

i<ω

contains Z2∞

odd even

Joshua Wiscons

Geometries and small groups of fMr

Algebraicity Conjecture Algebraicity Conjecture (Cherlin-Zilber) An infinite simple group of finite Morley rank is isomorphic to an (affine) algebraic group over an algebraically closed field. Analysis of groups of fMr breaks into 4 types.

does NOT contain

L

contains

L

i<ω

i<ω

does NOT contain Z2∞

contains Z2∞

Z2

deg.

odd

Z2

even

mixed

Joshua Wiscons

Geometries and small groups of fMr

Algebraicity Conjecture Algebraicity Conjecture (Cherlin-Zilber) An infinite simple group of finite Morley rank is isomorphic to an (affine) algebraic group over an algebraically closed field. Analysis of groups of fMr breaks into 4 types.

does NOT contain

L

contains

L

i<ω

i<ω

does NOT contain Z2∞

contains Z2∞

Z2

deg.

odd

Z2

even

mixed

Theorem (Altınel-Borovik-Cherlin) There are no infinite simple groups of finite Morley rank of mixed type and those of even type are indeed algebraic. Joshua Wiscons

Geometries and small groups of fMr

Algebraicity Conjecture Groups of fMr Simple groups of fMr ? Affine algebraic groups

Theorem (Altınel-Borovik-Cherlin) There are no infinite simple groups of finite Morley rank of mixed type and those of even type are indeed algebraic. Joshua Wiscons

Geometries and small groups of fMr

Algebraicity Conjecture Groups of fMr Simple groups of fMr

Deg?

O dd? Affine algebraic groups

Theorem (Altınel-Borovik-Cherlin) There are no infinite simple groups of finite Morley rank of mixed type and those of even type are indeed algebraic. Joshua Wiscons

Geometries and small groups of fMr

Algebraicity Conjecture Groups of fMr Simple groups of fMr

Deg?

O dd? Affine algebraic groups

Theorem (Altınel-Borovik-Cherlin) There are no infinite simple groups of finite Morley rank of mixed type and those of even type are indeed algebraic. Joshua Wiscons

Geometries and small groups of fMr

Algebraicity Conjecture

Simple groups of fMr

Deg?

O dd?

Theorem (Altınel-Borovik-Cherlin) There are no infinite simple groups of finite Morley rank of mixed type and those of even type are indeed algebraic. Joshua Wiscons

Geometries and small groups of fMr

Algebraicity Conjecture

Simple groups of fMr

O dd?

Deg?

does NOT contain

L

i<ω

Z2

does NOT contain Z2∞

contains Z2∞

deg.

odd

Joshua Wiscons

Geometries and small groups of fMr

Algebraicity Conjecture

Simple groups of fMr

O dd?

Deg?

maximal m for which the group contains

does NOT contain

L

i<ω

Z2

L

m

does NOT contain Z2∞

contains Z2∞

deg.

odd

Joshua Wiscons

Geometries and small groups of fMr

Z 2∞

Algebraicity Conjecture

Simple groups of fMr

O dd?

Deg?

maximal m for which the group contains

does NOT contain

L

i<ω

Z2

L

m

m=0

contains Z2∞

deg.

odd

Joshua Wiscons

Geometries and small groups of fMr

Z 2∞

Algebraicity Conjecture

Simple groups of fMr

O dd?

Deg?

maximal m for which the group contains

does NOT contain

L

i<ω

Z2

L

m=0

m≥1

deg.

odd

Joshua Wiscons

Geometries and small groups of fMr

m

Z 2∞

Algebraicity Conjecture

Simple groups of fMr

maximal m for which the group contains

does NOT contain

L

i<ω

Z2

L

m=0

m≥1

deg.

odd

Joshua Wiscons

Geometries and small groups of fMr

m

Z 2∞

Algebraicity Conjecture

Simple groups of fMr m=0 m=1 m=2

.. .

maximal m for which the group contains

does NOT contain

L

i<ω

Z2

L

m=0

m≥1

deg.

odd

Joshua Wiscons

Geometries and small groups of fMr

m

Z 2∞

Algebraicity Conjecture

Simple groups of fMr m=0 m=1 m=2

.. .

m=1 m=2

.. .

maximal m for which the group contains

does NOT contain

L

i<ω

Z2

L

m=0

m≥1

deg.

odd

Joshua Wiscons

Geometries and small groups of fMr

m

Z 2∞

Algebraicity Conjecture

Simple groups of fMr m=0 m=1 m=2

.. .

m=1

PSL2 (K)

m=2

.. .

maximal m for which the group contains

does NOT contain

L

i<ω

Z2

L

m=0

m≥1

deg.

odd

Joshua Wiscons

Geometries and small groups of fMr

m

Z 2∞

Algebraicity Conjecture

Simple groups of fMr m=0 m=1 m=2

.. .

m=1

PSL2 (K)

m=2

.. .

Today

Joshua Wiscons

Geometries and small groups of fMr

Algebraicity Conjecture

Simple groups of fMr m=0 m=1 m=2

.. .

m=1

PSL2 (K)

m=2

.. .

Today What: analyze “small” nonalgebraic configurations (which have m = 1, 2)

Joshua Wiscons

Geometries and small groups of fMr

Algebraicity Conjecture

Simple groups of fMr m=0 m=1 m=2

.. .

m=1

PSL2 (K)

m=2

.. .

Today What: analyze “small” nonalgebraic configurations (which have m = 1, 2) How: geometry of involutions

Joshua Wiscons

Geometries and small groups of fMr

Algebraicity Conjecture

Simple groups of fMr m=0 m=1 m=2

.. .

m=1

PSL2 (K)

m=2

.. .

Today What: analyze “small” nonalgebraic configurations (which have m = 1, 2) How: geometry of involutions Why: it works. . . Joshua Wiscons

Geometries and small groups of fMr

Notions of smallness We will consider the following notions of smallness.

Joshua Wiscons

Geometries and small groups of fMr

Notions of smallness We will consider the following notions of smallness. 1

Small Prüfer 2-rank, e.g. m ≤ 2

Joshua Wiscons

Geometries and small groups of fMr

Notions of smallness We will consider the following notions of smallness. 1

Small Prüfer 2-rank, e.g. m ≤ 2

2

Small amount of interesting (e.g. nonsolvable) subgroups

Joshua Wiscons

Geometries and small groups of fMr

Notions of smallness We will consider the following notions of smallness. 1

Small Prüfer 2-rank, e.g. m ≤ 2

2

Small amount of interesting (e.g. nonsolvable) subgroups

3

Small Morley rank

Joshua Wiscons

Geometries and small groups of fMr

Notions of smallness We will consider the following notions of smallness. 1

Small Prüfer 2-rank, e.g. m ≤ 2

2

Small amount of interesting (e.g. nonsolvable) subgroups

3

Small Morley rank

Joshua Wiscons

Geometries and small groups of fMr

Notions of smallness We will consider the following notions of smallness. 1

Small Prüfer 2-rank, e.g. m ≤ 2

2

Small amount of interesting (e.g. nonsolvable) subgroups

3

Small Morley rank

Definition A group G of fMr is an N-group if NG◦ (A) remains solvable whenever A is an infinite, definable, connected, and solvable subgroup of G.

Joshua Wiscons

Geometries and small groups of fMr

Notions of smallness We will consider the following notions of smallness. 1

Small Prüfer 2-rank, e.g. m ≤ 2

2

Small amount of interesting (e.g. nonsolvable) subgroups

3

Small Morley rank

Definition A group G of fMr is an N-group if NG◦ (A) remains solvable whenever A is an infinite, definable, connected, and solvable subgroup of G. Example A connected, algebraic N-group is either, solvable, SL2 (K), or PSL2 (K).

Joshua Wiscons

Geometries and small groups of fMr

Notions of smallness We will consider the following notions of smallness. 1

Small Prüfer 2-rank, e.g. m ≤ 2

2

Small amount of interesting (e.g. nonsolvable) subgroups

3

Small Morley rank

Definition A group G of fMr is an N-group if NG◦ (A) remains solvable whenever A is an infinite, definable, connected, and solvable subgroup of G. Example A connected, algebraic N-group is either, solvable, SL2 (K), or PSL2 (K). Theorem (Deloro-Jaligot ’16) An N-group has Prüfer 2-rank at most 2. Joshua Wiscons

Geometries and small groups of fMr

Notions of smallness

Simple groups of fMr m=0 m=1 m=2

.. .

m=1

PSL2 (K)

m=2

.. .

Theorem (Deloro-Jaligot ’16) An N-group has Prüfer 2-rank at most 2. Joshua Wiscons

Geometries and small groups of fMr

Notions of smallness

Simple groups of fMr m=0 m=1 m=2

N-groups

.. .

m=1

PSL2 (K)

m=2

.. .

Theorem (Deloro-Jaligot ’16) An N-group has Prüfer 2-rank at most 2. Joshua Wiscons

Geometries and small groups of fMr

Smallest of the small

Joshua Wiscons

Geometries and small groups of fMr

Smallest of the small Definition A group G of fMr is an N-group if NG◦ (A) remains solvable whenever A is an infinite, definable, connected, and solvable subgroup of G.

Joshua Wiscons

Geometries and small groups of fMr

Smallest of the small Definition A group G of fMr is an N-group if NG◦ (A) remains solvable whenever A is an infinite, definable, connected, and solvable subgroup of G. Definition An infinite simple group G of fMr is called minimal simple if every proper definable connected subgroup is solvable.

Joshua Wiscons

Geometries and small groups of fMr

Smallest of the small Definition A group G of fMr is an N-group if NG◦ (A) remains solvable whenever A is an infinite, definable, connected, and solvable subgroup of G. Definition An infinite simple group G of fMr is called minimal simple if every proper definable connected subgroup is solvable.

Remark minimal simple groups ⊆ simple N-groups Joshua Wiscons

Geometries and small groups of fMr

Smallest of the small Definition A group G of fMr is an N-group if NG◦ (A) remains solvable whenever A is an infinite, definable, connected, and solvable subgroup of G. Definition An infinite simple group G of fMr is called minimal simple if every proper definable connected subgroup is solvable. Definition An infinite simple group G of fMr is called bad if every proper definable connected subgroup is nilpotent. Remark minimal simple groups ⊆ simple N-groups Joshua Wiscons

Geometries and small groups of fMr

Smallest of the small Definition A group G of fMr is an N-group if NG◦ (A) remains solvable whenever A is an infinite, definable, connected, and solvable subgroup of G. Definition An infinite simple group G of fMr is called minimal simple if every proper definable connected subgroup is solvable. Definition An infinite simple group G of fMr is called bad if every proper definable connected subgroup is nilpotent. Remark simple bad groups ⊂ minimal simple groups ⊆ simple N-groups Joshua Wiscons

Geometries and small groups of fMr

Act II Geometry of Involutions

Joshua Wiscons

Geometries and small groups of fMr

A Theorem of Bachmann Theorem (Bachmann ’59 - reformulated by Schröder ’82) If the set I of involutions of a group G possess the the structure of a projective plane in such a way that three involutions are collinear iff their product is an involution, then hIi ∼ = SO3 (K, f ) for some interpretable K and anisotropic form f on K 3 .

Joshua Wiscons

Geometries and small groups of fMr

A Theorem of Bachmann Theorem (Bachmann ’59 - reformulated by Schröder ’82) If the set I of involutions of a group G possess the the structure of a projective plane in such a way that three involutions are collinear iff their product is an involution, then hIi ∼ = SO3 (K, f ) for some interpretable K and anisotropic form f on K 3 . Corollary No groups of fMr satisfy the hypotheses of Bachmann’s Theorem.

Joshua Wiscons

Geometries and small groups of fMr

A Theorem of Bachmann Theorem (Bachmann ’59 - reformulated by Schröder ’82) If the set I of involutions of a group G possess the the structure of a projective plane in such a way that three involutions are collinear iff their product is an involution, then hIi ∼ = SO3 (K, f ) for some interpretable K and anisotropic form f on K 3 . Corollary

Joshua Wiscons

Geometries and small groups of fMr

A Theorem of Bachmann Theorem (Bachmann ’59 - reformulated by Schröder ’82) If the set I of involutions of a group G possess the the structure of a projective plane in such a way that three involutions are collinear iff their product is an involution, then hIi ∼ = SO3 (K, f ) for some interpretable K and anisotropic form f on K 3 . Corollary For a group of fMr, the geometry of involutions with respect to B-collinearity can not be that of a projective plane.

Joshua Wiscons

Geometries and small groups of fMr

A Theorem of Bachmann Theorem (Bachmann ’59 - reformulated by Schröder ’82) If the set I of involutions of a group G possess the the structure of a projective plane in such a way that three involutions are collinear iff their product is an involution, then hIi ∼ = SO3 (K, f ) for some interpretable K and anisotropic form f on K 3 . Corollary For a group of fMr, the geometry of involutions with respect to B-collinearity can not be that of a projective plane. Definition (B-lines) We say that three involutions i, j, k are B-collinear if ijk is an involution.

Joshua Wiscons

Geometries and small groups of fMr

A Theorem of Bachmann

Corollary For a group of fMr, the geometry of involutions with respect to B-collinearity can not be that of a projective plane. Definition (B-lines) We say that three involutions i, j, k are B-collinear if ijk is an involution.

Joshua Wiscons

Geometries and small groups of fMr

A Theorem of Bachmann Corollary For a group of fMr, the geometry of involutions with respect to B-collinearity can not be that of a projective plane. Definition (B-lines) We say that three involutions i, j, k are B-collinear if ijk is an involution.

Joshua Wiscons

Geometries and small groups of fMr

A Theorem of Bachmann Corollary For a group of fMr, the geometry of involutions with respect to B-collinearity can not be that of a projective plane.

Joshua Wiscons

Geometries and small groups of fMr

A Theorem of Bachmann Corollary For a group of fMr, the geometry of involutions with respect to B-collinearity can not be that of a projective plane. A proof of the corollary without using Bachmann was given by the BN-pair.

Joshua Wiscons

Geometries and small groups of fMr

A Theorem of Bachmann Corollary For a group of fMr, the geometry of involutions with respect to B-collinearity can not be that of a projective plane. A proof of the corollary without using Bachmann was given by the BN-pair. Idea: embed the plane I in a 3-dimensional (Desarguesian) space.

Joshua Wiscons

Geometries and small groups of fMr

A Theorem of Bachmann Corollary For a group of fMr, the geometry of involutions with respect to B-collinearity can not be that of a projective plane. A proof of the corollary without using Bachmann was given by the BN-pair. Idea: embed the plane I in a 3-dimensional (Desarguesian) space. Points: elements of G

Joshua Wiscons

Geometries and small groups of fMr

A Theorem of Bachmann Corollary For a group of fMr, the geometry of involutions with respect to B-collinearity can not be that of a projective plane. A proof of the corollary without using Bachmann was given by the BN-pair. Idea: embed the plane I in a 3-dimensional (Desarguesian) space. Points: elements of G Lines: G-translates, `g, of lines from the plane associated to I

Joshua Wiscons

Geometries and small groups of fMr

A Theorem of Bachmann Corollary For a group of fMr, the geometry of involutions with respect to B-collinearity can not be that of a projective plane. A proof of the corollary without using Bachmann was given by the BN-pair. Idea: embed the plane I in a 3-dimensional (Desarguesian) space. Points: elements of G Lines: G-translates, `g, of lines from the plane associated to I Planes: G-translates, Ig, of the plane I

Joshua Wiscons

Geometries and small groups of fMr

A Theorem of Bachmann Corollary For a group of fMr, the geometry of involutions with respect to B-collinearity can not be that of a projective plane. A proof of the corollary without using Bachmann was given by the BN-pair. Idea: embed the plane I in a 3-dimensional (Desarguesian) space. Points: elements of G Lines: G-translates, `g, of lines from the plane associated to I Planes: G-translates, Ig, of the plane I

Note: G acts regularly on the points of this space.

Joshua Wiscons

Geometries and small groups of fMr

A Theorem of Bachmann Corollary For a group of fMr, the geometry of involutions with respect to B-collinearity can not be that of a projective plane. A proof of the corollary without using Bachmann was given by the BN-pair. Idea: embed the plane I in a 3-dimensional (Desarguesian) space. Points: elements of G Lines: G-translates, `g, of lines from the plane associated to I Planes: G-translates, Ig, of the plane I

Note: G acts regularly on the points of this space. Push a little more.

Joshua Wiscons

Geometries and small groups of fMr

A Theorem of Bachmann Corollary For a group of fMr, the geometry of involutions with respect to B-collinearity can not be that of a projective plane. A proof of the corollary without using Bachmann was given by the BN-pair. Idea: embed the plane I in a 3-dimensional (Desarguesian) space. Points: elements of G Lines: G-translates, `g, of lines from the plane associated to I Planes: G-translates, Ig, of the plane I

Note: G acts regularly on the points of this space. Push a little more. Done.

Joshua Wiscons

Geometries and small groups of fMr

A Theorem of Bachmann Corollary For a group of fMr, the geometry of involutions with respect to B-collinearity can not be that of a projective plane. A proof of the corollary without using Bachmann was given by the BN-pair. Idea: embed the plane I in a 3-dimensional (Desarguesian) space. Points: elements of G Lines: G-translates, `g, of lines from the plane associated to I Planes: G-translates, Ig, of the plane I

Note: G acts regularly on the points of this space. Push a little more. Done. Remark There is an important step not represented in the main idea above.

Joshua Wiscons

Geometries and small groups of fMr

A Theorem of Bachmann Corollary For a group of fMr, the geometry of involutions with respect to B-collinearity can not be that of a projective plane. A proof of the corollary without using Bachmann was given by the BN-pair. Idea: embed the plane I in a 3-dimensional (Desarguesian) space. Points: elements of G Lines: G-translates, `g, of lines from the plane associated to I Planes: G-translates, Ig, of the plane I

Note: G acts regularly on the points of this space. Push a little more. Done. Remark There is an important step not represented in the main idea above. Using that I is a projective plane with respect to B-collinearity, they show that B-collinearity is equivalent to C-collinearity. Joshua Wiscons

Geometries and small groups of fMr

B-Lines and C-Lines Definition (B-lines) We say that three involutions i, j, k are B-collinear if ijk is an involution.

Joshua Wiscons

Geometries and small groups of fMr

B-Lines and C-Lines Definition (B-lines) We say that three involutions i, j, k are B-collinear if ijk is an involution. Definition (C-lines) We define a C-line to be a set of the form CI (i) − {i} for i ∈ I.

Joshua Wiscons

Geometries and small groups of fMr

B-Lines and C-Lines Definition (B-lines) We say that three involutions i, j, k are B-collinear if ijk is an involution. Definition (C-lines) We define a C-line to be a set of the form CI (i) − {i} for i ∈ I. Remark

Joshua Wiscons

Geometries and small groups of fMr

B-Lines and C-Lines Definition (B-lines) We say that three involutions i, j, k are B-collinear if ijk is an involution. Definition (C-lines) We define a C-line to be a set of the form CI (i) − {i} for i ∈ I. Remark The geometry of C-lines is often easier to investigate.

Joshua Wiscons

Geometries and small groups of fMr

B-Lines and C-Lines Definition (B-lines) We say that three involutions i, j, k are B-collinear if ijk is an involution. Definition (C-lines) We define a C-line to be a set of the form CI (i) − {i} for i ∈ I. Remark The geometry of C-lines is often easier to investigate. Under favorable conditions, there is a built in polarity i ↔ CI (i) − {i}.

Joshua Wiscons

Geometries and small groups of fMr

B-Lines and C-Lines Definition (B-lines) We say that three involutions i, j, k are B-collinear if ijk is an involution. Definition (C-lines) We define a C-line to be a set of the form CI (i) − {i} for i ∈ I. Remark The geometry of C-lines is often easier to investigate. Under favorable conditions, there is a built in polarity i ↔ CI (i) − {i}. As mentioned before, when B-collinearity gives rise to a projective plane, B-lines and C-lines coincide.

Joshua Wiscons

Geometries and small groups of fMr

B-Lines and C-Lines Definition (B-lines) We say that three involutions i, j, k are B-collinear if ijk is an involution. Definition (C-lines) We define a C-line to be a set of the form CI (i) − {i} for i ∈ I. Remark The geometry of C-lines is often easier to investigate. Under favorable conditions, there is a built in polarity i ↔ CI (i) − {i}. As mentioned before, when B-collinearity gives rise to a projective plane, B-lines and C-lines coincide. Example If K is of fMr and char(K) 6= 2, then B-lines and C-lines coincide for PGL2 (K), but the geometry is not that of a projective plane. Joshua Wiscons

Geometries and small groups of fMr

B-Lines and C-Lines Definition (B-lines) We say that three involutions i, j, k are B-collinear if ijk is an involution. Definition (C-lines) We define a C-line to be a set of the form CI (i) − {i} for i ∈ I. Remark The geometry of C-lines is often easier to investigate. Under favorable conditions, there is a built in polarity i ↔ CI (i) − {i}. As mentioned before, when B-collinearity gives rise to a projective plane, B-lines and C-lines coincide.

Joshua Wiscons

Geometries and small groups of fMr

B-Lines and C-Lines Definition (B-lines) We say that three involutions i, j, k are B-collinear if ijk is an involution. Definition (C-lines) We define a C-line to be a set of the form CI (i) − {i} for i ∈ I. Remark The geometry of C-lines is often easier to investigate. Under favorable conditions, there is a built in polarity i ↔ CI (i) − {i}. As mentioned before, when B-collinearity gives rise to a projective plane, B-lines and C-lines coincide. Example If char(K) 6= 2, then K + o K × has no C-lines, but it does have (exactly) one B-line. Joshua Wiscons

Geometries and small groups of fMr

B-Lines and C-Lines Definition (B-lines) We say that three involutions i, j, k are B-collinear if ijk is an involution. Definition (C-lines) We define a C-line to be a set of the form CI (i) − {i} for i ∈ I. Remark The geometry of C-lines is often easier to investigate. Under favorable conditions, there is a built in polarity i ↔ CI (i) − {i}. As mentioned before, when B-collinearity gives rise to a projective plane, B-lines and C-lines coincide. Example If char(K) 6= 2, then K + o K × has no C-lines, but it does have (exactly) one B-line. (One needs commuting involutions to have C-lines.) Joshua Wiscons

Geometries and small groups of fMr

An (Old) Application

Simple groups of fMr m=0 m=1 m=2

N-groups

.. .

m=1

PSL2 (K)

m=2

.. .

Question

Joshua Wiscons

Geometries and small groups of fMr

An (Old) Application

Simple groups of fMr m=0 m=1 m=2

N-groups

.. .

m=1

PSL2 (K)

m=2

.. .

Question Recall: simple bad groups ⊂ minimal simple groups ⊆ simple N-groups.

Joshua Wiscons

Geometries and small groups of fMr

An (Old) Application

Simple groups of fMr m=0 m=1 m=2

N-groups

.. .

m=1

PSL2 (K)

m=2

.. .

Question Recall: simple bad groups ⊂ minimal simple groups ⊆ simple N-groups. Where do bad groups live?

Joshua Wiscons

Geometries and small groups of fMr

An (Old) Application

Simple groups of fMr m=0 m=1 m=2

Bad groups

N-groups

.. .

m=1

PSL2 (K)

m=2

.. .

Question Recall: simple bad groups ⊂ minimal simple groups ⊆ simple N-groups. Where do bad groups live?

Joshua Wiscons

Geometries and small groups of fMr

Bad Groups Theorem (Borovik ’83, Nesin ’89; Borovik-Poizat ’90, Corredor ’89) Bad groups have no involutions.

Joshua Wiscons

Geometries and small groups of fMr

Bad Groups Theorem (Borovik ’83, Nesin ’89; Borovik-Poizat ’90, Corredor ’89) Bad groups have no involutions. Proof Idea:

Joshua Wiscons

Geometries and small groups of fMr

Bad Groups Theorem (Borovik ’83, Nesin ’89; Borovik-Poizat ’90, Corredor ’89) Bad groups have no involutions. Proof Idea: 1

Assume involutions exist, and study C-lines.

Joshua Wiscons

Geometries and small groups of fMr

Bad Groups Theorem (Borovik ’83, Nesin ’89; Borovik-Poizat ’90, Corredor ’89) Bad groups have no involutions. Proof Idea: 1

Assume involutions exist, and study C-lines.

2

Show that the geometry of involutions with respect to C-collinearity is that of a projective plane.

Joshua Wiscons

Geometries and small groups of fMr

Bad Groups Theorem (Borovik ’83, Nesin ’89; Borovik-Poizat ’90, Corredor ’89) Bad groups have no involutions. Proof Idea: 1

Assume involutions exist, and study C-lines.

2

Show that the geometry of involutions with respect to C-collinearity is that of a projective plane.

3

Show C-lines coincide with B-lines.

Joshua Wiscons

Geometries and small groups of fMr

Bad Groups Theorem (Borovik ’83, Nesin ’89; Borovik-Poizat ’90, Corredor ’89) Bad groups have no involutions. Proof Idea: 1

Assume involutions exist, and study C-lines.

2

Show that the geometry of involutions with respect to C-collinearity is that of a projective plane.

3

Show C-lines coincide with B-lines.

4

Invoke Bachmann.

Joshua Wiscons

Geometries and small groups of fMr

Bad Groups Theorem (Borovik ’83, Nesin ’89; Borovik-Poizat ’90, Corredor ’89) Bad groups have no involutions. Proof Idea: 1

Assume involutions exist, and study C-lines.

2

Show that the geometry of involutions with respect to C-collinearity is that of a projective plane.

3

Show C-lines coincide with B-lines.

4

Invoke Bachmann. Done.

Joshua Wiscons

Geometries and small groups of fMr

Act III Applications to N-groups - 1 A small rank case study

Joshua Wiscons

Geometries and small groups of fMr

Analyzing N-groups Simple groups of fMr m=0 m=1 m=2

Bad groups

N-groups

.. .

m=1

PSL2 (K)

m=2

.. .

Joshua Wiscons

Geometries and small groups of fMr

Analyzing N-groups Simple groups of fMr m=0 m=1 m=2

Bad groups

N-groups

.. .

m=1

PSL2 (K)

m=2

.. .

Theorem (Deloro-Jaligot ’16) Let G be an infinite simple N-group of finite Morley rank.

Joshua Wiscons

Geometries and small groups of fMr

Analyzing N-groups Simple groups of fMr m=0 m=1 m=2

Bad groups

N-groups

.. .

m=1

PSL2 (K)

m=2

.. .

Theorem (Deloro-Jaligot ’16) Let G be an infinite simple N-group of finite Morley rank. Further assume that C◦ (i) is solvable for all i ∈ I(G).

Joshua Wiscons

Geometries and small groups of fMr

Analyzing N-groups Simple groups of fMr m=0 m=1 m=2

Bad groups

N-groups

.. .

m=1

PSL2 (K)

m=2

.. .

Theorem (Deloro-Jaligot ’16) Let G be an infinite simple N-group of finite Morley rank. Further assume that C◦ (i) is solvable for all i ∈ I(G). Then one of the following hold:

Joshua Wiscons

Geometries and small groups of fMr

Analyzing N-groups Simple groups of fMr m=0 m=1 m=2

Bad groups

N-groups

.. .

m=1

PSL2 (K)

m=2

.. .

Theorem (Deloro-Jaligot ’16) Let G be an infinite simple N-group of finite Morley rank. Further assume that C◦ (i) is solvable for all i ∈ I(G). Then one of the following hold: Deg. (m = 0):

G has no involutions

Joshua Wiscons

Geometries and small groups of fMr

Analyzing N-groups Simple groups of fMr m=0 m=1 m=2

Bad groups

N-groups

.. .

m=1

PSL2 (K)

m=2

.. .

Theorem (Deloro-Jaligot ’16) Let G be an infinite simple N-group of finite Morley rank. Further assume that C◦ (i) is solvable for all i ∈ I(G). Then one of the following hold: Deg. (m = 0): CiBo1 (m = 1):

G has no involutions C(i) = C◦ (i) (and S = Z2∞ )

Joshua Wiscons

Geometries and small groups of fMr

Analyzing N-groups Simple groups of fMr m=0 m=1 m=2

Bad groups

N-groups

.. .

m=1

PSL2 (K)

m=2

.. .

Theorem (Deloro-Jaligot ’16) Let G be an infinite simple N-group of finite Morley rank. Further assume that C◦ (i) is solvable for all i ∈ I(G). Then one of the following hold: Deg. (m = 0):

G has no involutions

CiBo1 (m = 1):

C(i) = C◦ (i) (and S = Z2∞ )

CiBo2 (m = 1):

C◦ (i) is inverted by any ω ∈ C(i) − {i} (and S = Z2∞ o hωi)

Joshua Wiscons

Geometries and small groups of fMr

Analyzing N-groups Simple groups of fMr m=0 m=1 m=2

Bad groups

N-groups

.. .

m=1

PSL2 (K)

m=2

.. .

Theorem (Deloro-Jaligot ’16) Let G be an infinite simple N-group of finite Morley rank. Further assume that C◦ (i) is solvable for all i ∈ I(G). Then one of the following hold: Deg. (m = 0):

G has no involutions

CiBo1 (m = 1):

C(i) = C◦ (i) (and S = Z2∞ )

CiBo2 (m = 1):

C◦ (i) is inverted by any ω ∈ C(i) − {i} (and S = Z2∞ o hωi)

CiBo3 (m = 2):

C(i) = C◦ (i) (and S = Z2∞ ⊕ Z2∞ )

Joshua Wiscons

Geometries and small groups of fMr

Analyzing N-groups Simple groups of fMr m=0 m=1 m=2

Bad groups

N-groups

.. .

m=1

PSL2 (K)

m=2

.. .

Theorem (Deloro-Jaligot ’16) Let G be an infinite simple N-group of finite Morley rank. Further assume that C◦ (i) is solvable for all i ∈ I(G). Then one of the following hold: Deg. (m = 0):

G has no involutions

CiBo1 (m = 1):

C(i) = C◦ (i) (and S = Z2∞ )

CiBo2 (m = 1):

C◦ (i) is inverted by any ω ∈ C(i) − {i} (and S = Z2∞ o hωi)

C(i) = C◦ (i) (and S = Z2∞ ⊕ Z2∞ ) Algebraic: G ∼ = PSL2 (K)

CiBo3 (m = 2):

Joshua Wiscons

Geometries and small groups of fMr

Analyzing N-groups

Theorem (Deloro-Jaligot ’16) Let G be an infinite simple N-group of finite Morley rank. Further assume that C◦ (i) is solvable for all i ∈ I(G). Then one of the following hold: Deg. (m = 0):

G has no involutions

CiBo1 (m = 1):

C(i) = C◦ (i) (and S = Z2∞ )

CiBo2 (m = 1):

C◦ (i) is inverted by any ω ∈ C(i) − {i} (and S = Z2∞ o hωi)

C(i) = C◦ (i) (and S = Z2∞ ⊕ Z2∞ ) Algebraic: G ∼ = PSL2 (K)

CiBo3 (m = 2):

Joshua Wiscons

Geometries and small groups of fMr

Analyzing N-groups Theorem (Deloro-Jaligot ’16) Let G be an infinite simple N-group of finite Morley rank. Further assume that C◦ (i) is solvable for all i ∈ I(G). Then one of the following hold: Deg. (m = 0):

G has no involutions

CiBo1 (m = 1):

C(i) = C◦ (i) (and S = Z2∞ )

CiBo2 (m = 1):

C◦ (i) is inverted by any ω ∈ C(i) − {i} (and S = Z2∞ o hωi)

C(i) = C◦ (i) (and S = Z2∞ ⊕ Z2∞ ) Algebraic: G ∼ = PSL2 (K)

CiBo3 (m = 2):

Joshua Wiscons

Geometries and small groups of fMr

Analyzing N-groups Theorem (Deloro-Jaligot ’16) Let G be an infinite simple N-group of finite Morley rank. Further assume that C◦ (i) is solvable for all i ∈ I(G). Then one of the following hold: Deg. (m = 0):

G has no involutions

CiBo1 (m = 1):

C(i) = C◦ (i) (and S = Z2∞ )

CiBo2 (m = 1):

C◦ (i) is inverted by any ω ∈ C(i) − {i} (and S = Z2∞ o hωi)

C(i) = C◦ (i) (and S = Z2∞ ⊕ Z2∞ ) Algebraic: G ∼ = PSL2 (K)

CiBo3 (m = 2):

The Goal

Joshua Wiscons

Geometries and small groups of fMr

Analyzing N-groups Theorem (Deloro-Jaligot ’16) Let G be an infinite simple N-group of finite Morley rank. Further assume that C◦ (i) is solvable for all i ∈ I(G). Then one of the following hold: Deg. (m = 0):

G has no involutions

CiBo1 (m = 1):

C(i) = C◦ (i) (and S = Z2∞ )

CiBo2 (m = 1):

C◦ (i) is inverted by any ω ∈ C(i) − {i} (and S = Z2∞ o hωi)

C(i) = C◦ (i) (and S = Z2∞ ⊕ Z2∞ ) Algebraic: G ∼ = PSL2 (K)

CiBo3 (m = 2):

The Goal What: As before: analyze “small” nonalgebraic configurations

Joshua Wiscons

Geometries and small groups of fMr

Analyzing N-groups Theorem (Deloro-Jaligot ’16) Let G be an infinite simple N-group of finite Morley rank. Further assume that C◦ (i) is solvable for all i ∈ I(G). Then one of the following hold: Deg. (m = 0):

G has no involutions

CiBo1 (m = 1):

C(i) = C◦ (i) (and S = Z2∞ )

CiBo2 (m = 1):

C◦ (i) is inverted by any ω ∈ C(i) − {i} (and S = Z2∞ o hωi)

C(i) = C◦ (i) (and S = Z2∞ ⊕ Z2∞ ) Algebraic: G ∼ = PSL2 (K)

CiBo3 (m = 2):

The Goal What:

Joshua Wiscons

Geometries and small groups of fMr

Analyzing N-groups Theorem (Deloro-Jaligot ’16) Let G be an infinite simple N-group of finite Morley rank. Further assume that C◦ (i) is solvable for all i ∈ I(G). Then one of the following hold: Deg. (m = 0):

G has no involutions

CiBo1 (m = 1):

C(i) = C◦ (i) (and S = Z2∞ )

CiBo2 (m = 1):

C◦ (i) is inverted by any ω ∈ C(i) − {i} (and S = Z2∞ o hωi)

C(i) = C◦ (i) (and S = Z2∞ ⊕ Z2∞ ) Algebraic: G ∼ = PSL2 (K)

CiBo3 (m = 2):

The Goal What: Better understand the CiBo configurations by analyzing them in a small rank setting

Joshua Wiscons

Geometries and small groups of fMr

Analyzing N-groups Theorem (Deloro-Jaligot ’16) Let G be an infinite simple N-group of finite Morley rank. Further assume that C◦ (i) is solvable for all i ∈ I(G). Then one of the following hold: Deg. (m = 0):

G has no involutions

CiBo1 (m = 1):

C(i) = C◦ (i) (and S = Z2∞ )

CiBo2 (m = 1):

C◦ (i) is inverted by any ω ∈ C(i) − {i} (and S = Z2∞ o hωi)

C(i) = C◦ (i) (and S = Z2∞ ⊕ Z2∞ ) Algebraic: G ∼ = PSL2 (K)

CiBo3 (m = 2):

The Goal What: Better understand the CiBo configurations by analyzing them in a small rank setting How: geometry of involutions

Joshua Wiscons

Geometries and small groups of fMr

Groups of small Morley rank Theorem Let G be a simple group of fMr.

Joshua Wiscons

Geometries and small groups of fMr

Groups of small Morley rank Theorem Let G be a simple group of fMr. 1

rk G = 1 =⇒ not possible

Joshua Wiscons

Geometries and small groups of fMr

Groups of small Morley rank Theorem Let G be a simple group of fMr. 1

rk G = 1 =⇒ not possible (G is abelian) [Reineke ’75]

Joshua Wiscons

Geometries and small groups of fMr

Groups of small Morley rank Theorem Let G be a simple group of fMr. 1

rk G = 1 =⇒ not possible (G is abelian) [Reineke ’75]

2

rk G = 2 =⇒ not possible (G is solvable) [Cherlin ’79]

Joshua Wiscons

Geometries and small groups of fMr

Groups of small Morley rank Theorem Let G be a simple group of fMr. 1

rk G = 1 =⇒ not possible (G is abelian) [Reineke ’75]

2

rk G = 2 =⇒ not possible (G is solvable) [Cherlin ’79] rk G = 3 =⇒

3

Joshua Wiscons

Geometries and small groups of fMr

Groups of small Morley rank Theorem Let G be a simple group of fMr. 1

rk G = 1 =⇒ not possible (G is abelian) [Reineke ’75]

2

rk G = 2 =⇒ not possible (G is solvable) [Cherlin ’79] rk G = 3 =⇒ G is PSL2 (K) or a bad group [Cherlin ’79]

3

Joshua Wiscons

Geometries and small groups of fMr

Groups of small Morley rank Theorem Let G be a simple group of fMr. 1

rk G = 1 =⇒ not possible (G is abelian) [Reineke ’75]

2

rk G = 2 =⇒ not possible (G is solvable) [Cherlin ’79] rk G = 3 =⇒ G is PSL2 (K) or a bad group [Cherlin ’79]

3

Geometry of involutions shows bad groups have no involutions.

Joshua Wiscons

Geometries and small groups of fMr

Groups of small Morley rank Theorem Let G be a simple group of fMr. 1

rk G = 1 =⇒ not possible (G is abelian) [Reineke ’75]

2

rk G = 2 =⇒ not possible (G is solvable) [Cherlin ’79] [Frécon ’16] rk G = 3 =⇒ G is PSL2 (K) or a bad group [Cherlin ’79]

3

Geometry of involutions shows bad groups have no involutions.

Joshua Wiscons

Geometries and small groups of fMr

Groups of small Morley rank Theorem Let G be a simple group of fMr. 1

rk G = 1 =⇒ not possible (G is abelian) [Reineke ’75]

2

rk G = 2 =⇒ not possible (G is solvable) [Cherlin ’79] [Frécon ’16] rk G = 3 =⇒ G is PSL2 (K) or a bad group [Cherlin ’79]

3

Geometry of involutions shows bad groups have no involutions. 4

rk G = 4 =⇒

Joshua Wiscons

Geometries and small groups of fMr

Groups of small Morley rank Theorem Let G be a simple group of fMr. 1

rk G = 1 =⇒ not possible (G is abelian) [Reineke ’75]

2

rk G = 2 =⇒ not possible (G is solvable) [Cherlin ’79] [Frécon ’16] rk G = 3 =⇒ G is PSL2 (K) or a bad group [Cherlin ’79]

3

Geometry of involutions shows bad groups have no involutions. 4

rk G = 4 =⇒ G is a bad group [W ’14]

Joshua Wiscons

Geometries and small groups of fMr

Groups of small Morley rank Theorem Let G be a simple group of fMr. 1

rk G = 1 =⇒ not possible (G is abelian) [Reineke ’75]

2

rk G = 2 =⇒ not possible (G is solvable) [Cherlin ’79] [Frécon ’16] rk G = 3 =⇒ G is PSL2 (K) or a bad group [Cherlin ’79]

3

Geometry of involutions shows bad groups have no involutions. 4

rk G = 4 =⇒ G is a bad group [W ’14] Geometry of involutions is used (with a different punchline) following Nesin’s work on sharply 2-transitive groups

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad.

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Proof Outline:

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Proof Outline: 1 Preliminary analysis

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Proof Outline: 1 Preliminary analysis ABC: G has degenerate or odd type

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Proof Outline: 1 Preliminary analysis ABC: G has degenerate or odd type Hrushovski: G has no subgroups of rank 4

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Proof Outline: 1 Preliminary analysis ABC: G has degenerate or odd type Hrushovski: G has no subgroups of rank 4 This implies that G is an N-group

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Proof Outline: 1 Preliminary analysis ABC: G has degenerate or odd type Hrushovski: G has no subgroups of rank 4 This implies that G is an N-group

Strong uniqueness principle: if B1 and B2 are Borels, then (F(B1 ) ∩ F(B2 ))◦ = 1

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Proof Outline: 1 Preliminary analysis ABC: G has degenerate or odd type Hrushovski: G has no subgroups of rank 4 This implies that G is an N-group

Strong uniqueness principle: if B1 and B2 are Borels, then (F(B1 ) ∩ F(B2 ))◦ = 1 2

C◦ (i) is solvable, so Deloro-Jaligot analysis applies

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Proof Outline: 1 Preliminary analysis ABC: G has degenerate or odd type Hrushovski: G has no subgroups of rank 4 This implies that G is an N-group

Strong uniqueness principle: if B1 and B2 are Borels, then (F(B1 ) ∩ F(B2 ))◦ = 1 2

C◦ (i) is solvable, so Deloro-Jaligot analysis applies

3

If G has degenerate type, it is bad

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Proof Outline: 1 Preliminary analysis ABC: G has degenerate or odd type Hrushovski: G has no subgroups of rank 4 This implies that G is an N-group

Strong uniqueness principle: if B1 and B2 are Borels, then (F(B1 ) ∩ F(B2 ))◦ = 1 2

C◦ (i) is solvable, so Deloro-Jaligot analysis applies

3

If G has degenerate type, it is bad

4

Thus G has odd type, and we have three CiBo cases to consider

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo3 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo3

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo3 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo3 (i) Basic observations

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo3 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo3 (i) Basic observations Recall: C(i) = C◦ (i) and S = Z2∞ ⊕ Z2∞

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo3 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo3 (i) Basic observations Recall: C(i) = C◦ (i) and S = Z2∞ ⊕ Z2∞ Also by Deloro-Jaligot, i 6= j =⇒ C(i) 6= C(j)

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo3 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo3 (i) Basic observations Recall: C(i) = C◦ (i) and S = Z2∞ ⊕ Z2∞ Also by Deloro-Jaligot, i 6= j =⇒ C(i) 6= C(j) C(i) has rank 3, and F(C(i)) has rank 2

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo3 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo3 (i) Basic observations Recall: C(i) = C◦ (i) and S = Z2∞ ⊕ Z2∞ Also by Deloro-Jaligot, i 6= j =⇒ C(i) 6= C(j) C(i) has rank 3, and F(C(i)) has rank 2 C(i)/F(C(i)) contains a 2-torus

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo3 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo3 (i) Basic observations Recall: C(i) = C◦ (i) and S = Z2∞ ⊕ Z2∞ Also by Deloro-Jaligot, i 6= j =⇒ C(i) 6= C(j) C(i) has rank 3, and F(C(i)) has rank 2 C(i)/F(C(i)) contains a 2-torus

(ii) C(i, j) contains a unique involution distinct from i and j

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo3 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo3 (i) Basic observations Recall: C(i) = C◦ (i) and S = Z2∞ ⊕ Z2∞ Also by Deloro-Jaligot, i 6= j =⇒ C(i) 6= C(j) C(i) has rank 3, and F(C(i)) has rank 2 C(i)/F(C(i)) contains a 2-torus

(ii) C(i, j) contains a unique involution distinct from i and j Rank considerations force C(i, j) to be infinite

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo3 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo3 (i) Basic observations Recall: C(i) = C◦ (i) and S = Z2∞ ⊕ Z2∞ Also by Deloro-Jaligot, i 6= j =⇒ C(i) 6= C(j) C(i) has rank 3, and F(C(i)) has rank 2 C(i)/F(C(i)) contains a 2-torus

(ii) C(i, j) contains a unique involution distinct from i and j Rank considerations force C(i, j) to be infinite “Uniqueness” forces C(i, j) to be abelian and contain a 2-torus

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo3 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo3 (i) Basic observations Recall: C(i) = C◦ (i) and S = Z2∞ ⊕ Z2∞ Also by Deloro-Jaligot, i 6= j =⇒ C(i) 6= C(j) C(i) has rank 3, and F(C(i)) has rank 2 C(i)/F(C(i)) contains a 2-torus

(ii) C(i, j) contains a unique involution distinct from i and j Rank considerations force C(i, j) to be infinite “Uniqueness” forces C(i, j) to be abelian and contain a 2-torus If pr2 C◦ (i, j) = 2, then I(C(i, j)) = {i, j, ij}

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo3 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo3 (i) Basic observations Recall: C(i) = C◦ (i) and S = Z2∞ ⊕ Z2∞ Also by Deloro-Jaligot, i 6= j =⇒ C(i) 6= C(j) C(i) has rank 3, and F(C(i)) has rank 2 C(i)/F(C(i)) contains a 2-torus

(ii) C(i, j) contains a unique involution distinct from i and j Rank considerations force C(i, j) to be infinite “Uniqueness” forces C(i, j) to be abelian and contain a 2-torus If pr2 C◦ (i, j) = 2, then I(C(i, j)) = {i, j, ij} If pr2 C◦ (i, j) = 1, then for k ∈ I(C◦ (i, j)), {k} ⊆ I(C(i, j)) ⊆ {i, j, k}

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo3 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo3 (i) Basic observations Recall: C(i) = C◦ (i) and S = Z2∞ ⊕ Z2∞ Also by Deloro-Jaligot, i 6= j =⇒ C(i) 6= C(j) C(i) has rank 3, and F(C(i)) has rank 2 C(i)/F(C(i)) contains a 2-torus

(ii) C(i, j) contains a unique involution distinct from i and j

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo3 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo3 (i) Basic observations Recall: C(i) = C◦ (i) and S = Z2∞ ⊕ Z2∞ Also by Deloro-Jaligot, i 6= j =⇒ C(i) 6= C(j) C(i) has rank 3, and F(C(i)) has rank 2 C(i)/F(C(i)) contains a 2-torus

(ii) C(i, j) contains a unique involution distinct from i and j (iii) C-lines give rise to a projective plane!

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo3 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo3 (i) Basic observations Recall: C(i) = C◦ (i) and S = Z2∞ ⊕ Z2∞ Also by Deloro-Jaligot, i 6= j =⇒ C(i) 6= C(j) C(i) has rank 3, and F(C(i)) has rank 2 C(i)/F(C(i)) contains a 2-torus

(ii) C(i, j) contains a unique involution distinct from i and j (iii) C-lines give rise to a projective plane! Recall lines are of the form i⊥ := CI (i) − {i}

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo3 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo3 (i) Basic observations Recall: C(i) = C◦ (i) and S = Z2∞ ⊕ Z2∞ Also by Deloro-Jaligot, i 6= j =⇒ C(i) 6= C(j) C(i) has rank 3, and F(C(i)) has rank 2 C(i)/F(C(i)) contains a 2-torus

(ii) C(i, j) contains a unique involution distinct from i and j (iii) C-lines give rise to a projective plane! Recall lines are of the form i⊥ := CI (i) − {i} Two lines intersect in a unique point

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo3 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo3 (i) Basic observations Recall: C(i) = C◦ (i) and S = Z2∞ ⊕ Z2∞ Also by Deloro-Jaligot, i 6= j =⇒ C(i) 6= C(j) C(i) has rank 3, and F(C(i)) has rank 2 C(i)/F(C(i)) contains a 2-torus

(ii) C(i, j) contains a unique involution distinct from i and j (iii) C-lines give rise to a projective plane! Recall lines are of the form i⊥ := CI (i) − {i} Two lines intersect in a unique point Two points determine a unique line

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo3 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo3 (i) Basic observations Recall: C(i) = C◦ (i) and S = Z2∞ ⊕ Z2∞ Also by Deloro-Jaligot, i 6= j =⇒ C(i) 6= C(j) C(i) has rank 3, and F(C(i)) has rank 2 C(i)/F(C(i)) contains a 2-torus

(ii) C(i, j) contains a unique involution distinct from i and j (iii) C-lines give rise to a projective plane! Recall lines are of the form i⊥ := CI (i) − {i} Two lines intersect in a unique point Two points determine a unique line Four independent involutions have the property that no three are collinear

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo3 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo3 (i) Basic observations Recall: C(i) = C◦ (i) and S = Z2∞ ⊕ Z2∞ Also by Deloro-Jaligot, i 6= j =⇒ C(i) 6= C(j) C(i) has rank 3, and F(C(i)) has rank 2 C(i)/F(C(i)) contains a 2-torus

(ii) C(i, j) contains a unique involution distinct from i and j (iii) C-lines give rise to a projective plane! Recall lines are of the form i⊥ := CI (i) − {i} Two lines intersect in a unique point Two points determine a unique line Four independent involutions have the property that no three are collinear

(iv) C-lines coincide with B-lines. Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo3 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo3 (i) Basic observations Recall: C(i) = C◦ (i) and S = Z2∞ ⊕ Z2∞ Also by Deloro-Jaligot, i 6= j =⇒ C(i) 6= C(j) C(i) has rank 3, and F(C(i)) has rank 2 C(i)/F(C(i)) contains a 2-torus

(ii) C(i, j) contains a unique involution distinct from i and j (iii) C-lines give rise to a projective plane! Recall lines are of the form i⊥ := CI (i) − {i} Two lines intersect in a unique point Two points determine a unique line Four independent involutions have the property that no three are collinear

(iv) C-lines coincide with B-lines. “Bachmann.” Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo3 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo3 (i) Basic observations Recall: C(i) = C◦ (i) and S = Z2∞ ⊕ Z2∞ Also by Deloro-Jaligot, i 6= j =⇒ C(i) 6= C(j) C(i) has rank 3, and F(C(i)) has rank 2 C(i)/F(C(i)) contains a 2-torus

(ii) C(i, j) contains a unique involution distinct from i and j (iii) C-lines give rise to a projective plane! Recall lines are of the form i⊥ := CI (i) − {i} Two lines intersect in a unique point Two points determine a unique line Four independent involutions have the property that no three are collinear

(iv) C-lines coincide with B-lines. “Bachmann.” Done. Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo2 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo2

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo2 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo2 (i) Deloro-Jaligot has more to say about CiBo2 :

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo2 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo2 (i) Deloro-Jaligot has more to say about CiBo2 : rk G = 3 · rk C(i)

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo2 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo2 (i) Deloro-Jaligot has more to say about CiBo2 : rk G = 3 · rk C(i) (ii) Done.

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo2 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo2 (i) Deloro-Jaligot has more to say about CiBo2 : rk G = 3 · rk C(i) (ii) Done. And unsatisfying.

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo2 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo2 (i) Deloro-Jaligot has more to say about CiBo2 : rk G = 3 · rk C(i) (ii) Done. And unsatisfying. (More will be said about CiBo2 in a moment.)

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo1 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo1

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo1 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo1 (i) There exist a rank 3 Borel B with finite center (equal to C(i) or C◦ (ij)).

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo1 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo1 (i) There exist a rank 3 Borel B with finite center (equal to C(i) or C◦ (ij)). There are a few steps here, but the Brauer-Fowler map I × I 7→ G is key.

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo1 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo1 (i) There exist a rank 3 Borel B with finite center (equal to C(i) or C◦ (ij)). There are a few steps here, but the Brauer-Fowler map I × I 7→ G is key.

(ii) Geometry again

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo1 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Assume CiBo1 (i) There exist a rank 3 Borel B with finite center (equal to C(i) or C◦ (ij)). There are a few steps here, but the Brauer-Fowler map I × I 7→ G is key.

(ii) Geometry again. . . kind of

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo1 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. HOME

SPEAKERS

PHOTOS

PROGRAM

CONTACT

PARTICIPANTS

VENUE

TRAVEL

ACCOMMODATION

Assume CiBo1

SPONSORS

(i) There exist a rank 3 Borel BN-Pair B with finite center (equal to C(i) or C◦ (ij)). HOME

PHOTOS

SPEAKERS

CONTACT

PROGRAM

SPONSORS

PARTICIPANTS

VENUE

TRAVEL

ACCOMMODATION

Conference in honour of 60th birthdays of

There are a few steps here, but the Brauer-Fowler map I × I 7→ G is key. Alexandre Borovik and Ali Nesin

BN-Pair 19-22 October 2016, Istanbul

Conference in honour of 60th birthdays of Alexandre Borovik and Ali Nesin

(ii) Geometry again. . . kind of

19-22 October 2016, Istanbul

B is a part of a split

of (Tits) rank 1.

Organizers: Tuna Altınel (Lyon), Ayşe Berkman (İstanbul), Özlem Beyarslan (İstanbul),

Organizers: Tuna Altınel (Lyon), Ayşe Berkman (İstanbul), Özlem Beyarslan (İstanbul),

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo1 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. HOME

SPEAKERS

PHOTOS

PROGRAM

CONTACT

PARTICIPANTS

VENUE

TRAVEL

ACCOMMODATION

Assume CiBo1

SPONSORS

(i) There exist a rank 3 Borel BN-Pair B with finite center (equal to C(i) or C◦ (ij)). HOME

SPEAKERS

HOME SPEAKERS PROGRAM PARTICIPANTS VENUE PROGRAM PARTICIPANTS VENUE

PHOTOS

CONTACT

SPONSORS

ACCOMMODATION TRAVEL ACCOMMODATION

TRAVEL

Conference in honour of 60th birthdays of

There are a few steps here, but the Brauer-Fowler map I × I 7→ G is key. PHOTOS

CONTACT

Alexandre Borovik and Ali Nesin

SPONSORS

BN-Pair 19-22 October 2016, Istanbul

Conference in honour of 60th birthdays of Alexandre Borovik and Ali Nesin

(ii) Geometry again. . . kind of BN-Pair HOME

PHOTOS

SPEAKERS

CONTACT

PROGRAM

SPONSORS

PARTICIPANTS

VENUE

TRAVEL

ACCOMMODATION 19-22 October

2016, Istanbul

Conference in honour of 60th birthdays of Alexandre Borovik and Ali Nesin

BN-Pair 19-22 October 2016, Istanbul

B is a part of a split Conference in honour of 60th birthdays of Alexandre Borovik and Ali Nesin

of (Tits) rank 1.

19-22 October 2016, Istanbul

Organizers: Tuna Altınel (Lyon), Ayşe Berkman (İstanbul), Özlem Beyarslan (İstanbul),

In general, a

gives rise to a geometry known as a building. Organizers: Tuna Altınel (Lyon), Ayşe Berkman (İstanbul), Özlem Beyarslan (İstanbul),

Organizers: Tuna Altınel (Lyon), Ayşe Berkman (İstanbul), Özlem Beyarslan (İstanbul),

Organizers: Tuna Altınel (Lyon), Ayşe Berkman (İstanbul), Özlem Beyarslan (İstanbul),

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo1 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. HOME

SPEAKERS

PHOTOS

PROGRAM

CONTACT

PARTICIPANTS

VENUE

TRAVEL

ACCOMMODATION

Assume CiBo1

SPONSORS

(i) There exist a rank 3 Borel BN-Pair B with finite center (equal to C(i) or C◦ (ij)). HOME

SPEAKERS

HOME SPEAKERS PROGRAM PARTICIPANTS VENUE PROGRAM PARTICIPANTS VENUE

PHOTOS

CONTACT

SPONSORS

ACCOMMODATION TRAVEL ACCOMMODATION

TRAVEL

Conference in honour of 60th birthdays of

There are a few steps here, but the Brauer-Fowler map I × I 7→ G is key. PHOTOS

CONTACT

Alexandre Borovik and Ali Nesin

SPONSORS

BN-Pair 19-22 October 2016, Istanbul

Conference in honour of 60th birthdays of Alexandre Borovik and Ali Nesin

(ii) Geometry again. . . kind of BN-Pair HOME

PHOTOS

SPEAKERS

CONTACT

PROGRAM

SPONSORS

PARTICIPANTS

VENUE

TRAVEL

ACCOMMODATION 19-22 October

2016, Istanbul

Conference in honour of 60th birthdays of Alexandre Borovik and Ali Nesin

BN-Pair 19-22 October 2016, Istanbul

B is a part of a split Conference in honour of 60th birthdays of Alexandre Borovik and Ali Nesin

of (Tits) rank 1.

19-22 October 2016, Istanbul

Organizers: Tuna Altınel (Lyon), Ayşe Berkman (İstanbul), Özlem Beyarslan (İstanbul),

In general, a

gives rise to a geometry known as a building. Organizers: Tuna Altınel (Lyon), Ayşe Berkman (İstanbul), Özlem Beyarslan (İstanbul),

But in Tits rank 1, one obtains just a set on which G acts 2-transitively. Organizers: Tuna Altınel (Lyon), Ayşe Berkman (İstanbul), Özlem Beyarslan (İstanbul),

Organizers: Tuna Altınel (Lyon), Ayşe Berkman (İstanbul), Özlem Beyarslan (İstanbul),

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo1 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. HOME

SPEAKERS

PHOTOS

PROGRAM

CONTACT

PARTICIPANTS

VENUE

TRAVEL

ACCOMMODATION

Assume CiBo1

SPONSORS

(i) There exist a rank 3 Borel BN-Pair B with finite center (equal to C(i) or C◦ (ij)). HOME

SPEAKERS

HOME SPEAKERS PROGRAM PARTICIPANTS VENUE PROGRAM PARTICIPANTS VENUE

PHOTOS

CONTACT

SPONSORS

ACCOMMODATION TRAVEL ACCOMMODATION

TRAVEL

Conference in honour of 60th birthdays of

There are a few steps here, but the Brauer-Fowler map I × I 7→ G is key. PHOTOS

CONTACT

Alexandre Borovik and Ali Nesin

SPONSORS

BN-Pair 19-22 October 2016, Istanbul

Conference in honour of 60th birthdays of Alexandre Borovik and Ali Nesin

(ii) Geometry again. . . kind of BN-Pair HOME

PHOTOS

SPEAKERS

CONTACT

PROGRAM

SPONSORS

PARTICIPANTS

VENUE

TRAVEL

ACCOMMODATION 19-22 October

2016, Istanbul

Conference in honour of 60th birthdays of Alexandre Borovik and Ali Nesin

BN-Pair 19-22 October 2016, Istanbul

B is a part of a split Conference in honour of 60th birthdays of Alexandre Borovik and Ali Nesin

of (Tits) rank 1.

19-22 October 2016, Istanbul

Organizers: Tuna Altınel (Lyon), Ayşe Berkman (İstanbul), Özlem Beyarslan (İstanbul),

In general, a

gives rise to a geometry known as a building. Organizers: Tuna Altınel (Lyon), Ayşe Berkman (İstanbul), Özlem Beyarslan (İstanbul),

But in Tits rank 1, one obtains just a set on which G acts 2-transitively. In our case, we also have that the 2-point stabilizer is infinite and abelian. Organizers: Tuna Altınel (Lyon), Ayşe Berkman (İstanbul), Özlem Beyarslan (İstanbul),

Organizers: Tuna Altınel (Lyon), Ayşe Berkman (İstanbul), Özlem Beyarslan (İstanbul),

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo1 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. HOME

SPEAKERS

PHOTOS

PROGRAM

CONTACT

PARTICIPANTS

VENUE

TRAVEL

ACCOMMODATION

Assume CiBo1

SPONSORS

(i) There exist a rank 3 Borel BN-Pair B with finite center (equal to C(i) or C◦ (ij)). HOME

SPEAKERS

HOME SPEAKERS PROGRAM PARTICIPANTS VENUE PROGRAM PARTICIPANTS VENUE

PHOTOS

CONTACT

SPONSORS

ACCOMMODATION TRAVEL ACCOMMODATION

TRAVEL

Conference in honour of 60th birthdays of

There are a few steps here, but the Brauer-Fowler map I × I 7→ G is key. PHOTOS

CONTACT

Alexandre Borovik and Ali Nesin

SPONSORS

BN-Pair 19-22 October 2016, Istanbul

Conference in honour of 60th birthdays of Alexandre Borovik and Ali Nesin

(ii) Geometry again. . . kind of BN-Pair HOME

PHOTOS

SPEAKERS

CONTACT

PROGRAM

SPONSORS

PARTICIPANTS

VENUE

ACCOMMODATION 19-22 October

TRAVEL

2016, Istanbul

Conference in honour of 60th birthdays of Alexandre Borovik and Ali Nesin

BN-Pair 19-22 October 2016, Istanbul

B is a part of a split Conference in honour of 60th birthdays of Alexandre Borovik and Ali Nesin

HOME

SPEAKERS

PROGRAM

PARTICIPANTS

VENUE

of (Tits) rank 1. TRAVEL

ACCOMMODATION

19-22 October 2016, Istanbul

PHOTOS

CONTACT

SPONSORS Organizers:

Tuna Altınel (Lyon), Ayşe Berkman (İstanbul), Özlem Beyarslan (İstanbul),

In general, a HOME

PHOTOS

gives rise to a geometry known as a building.

SPEAKERS

CONTACT

PROGRAM

SPONSORS

PARTICIPANTS

VENUE

TRAVEL

BN-Pair ACCOMMODATION

Organizers: Tuna Altınel (Lyon), Ayşe Berkman (İstanbul), Özlem Beyarslan (İstanbul),

Conference in honour of 60th birthdays of Alexandre Borovik and Ali Nesin

But in Tits rank 1, one obtains just a set on which G acts 2-transitively. In our case, we also have that the 2-point stabilizer is infinite and abelian. Organizers: Tuna Altınel (Lyon), Ayşe Berkman (İstanbul), Özlem Beyarslan (İstanbul),

BN-Pair 19-22 October 2016, Istanbul

Conference in honour of 60th birthdays of Alexandre Borovik and Ali Nesin 19-22 October 2016, Istanbul

Organizers: Tuna Altınel (Lyon), Ayşe Berkman (İstanbul), Özlem Beyarslan (İstanbul),

of Tits rank 1, G is PSL2 (K).

(iii) By the theory of split Organizers: Tuna Altınel (Lyon), Ayşe Berkman (İstanbul), Özlem Beyarslan (İstanbul),

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo1 Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. HOME

SPEAKERS

PHOTOS

PROGRAM

CONTACT

PARTICIPANTS

VENUE

TRAVEL

ACCOMMODATION

Assume CiBo1

SPONSORS

(i) There exist a rank 3 Borel BN-Pair B with finite center (equal to C(i) or C◦ (ij)). HOME

SPEAKERS

HOME SPEAKERS PROGRAM PARTICIPANTS VENUE PROGRAM PARTICIPANTS VENUE

PHOTOS

CONTACT

SPONSORS

ACCOMMODATION TRAVEL ACCOMMODATION

TRAVEL

Conference in honour of 60th birthdays of

There are a few steps here, but the Brauer-Fowler map I × I 7→ G is key. PHOTOS

CONTACT

Alexandre Borovik and Ali Nesin

SPONSORS

BN-Pair 19-22 October 2016, Istanbul

Conference in honour of 60th birthdays of Alexandre Borovik and Ali Nesin

(ii) Geometry again. . . kind of BN-Pair HOME

PHOTOS

SPEAKERS

CONTACT

PROGRAM

SPONSORS

PARTICIPANTS

VENUE

ACCOMMODATION 19-22 October

TRAVEL

2016, Istanbul

Conference in honour of 60th birthdays of Alexandre Borovik and Ali Nesin

BN-Pair 19-22 October 2016, Istanbul

B is a part of a split Conference in honour of 60th birthdays of Alexandre Borovik and Ali Nesin

HOME

SPEAKERS

PROGRAM

PARTICIPANTS

VENUE

of (Tits) rank 1. TRAVEL

ACCOMMODATION

19-22 October 2016, Istanbul

PHOTOS

CONTACT

SPONSORS Organizers:

Tuna Altınel (Lyon), Ayşe Berkman (İstanbul), Özlem Beyarslan (İstanbul),

In general, a HOME

PHOTOS

gives rise to a geometry known as a building.

SPEAKERS

CONTACT

PROGRAM

SPONSORS

PARTICIPANTS

VENUE

TRAVEL

BN-Pair ACCOMMODATION

Organizers: Tuna Altınel (Lyon), Ayşe Berkman (İstanbul), Özlem Beyarslan (İstanbul),

Conference in honour of 60th birthdays of Alexandre Borovik and Ali Nesin

But in Tits rank 1, one obtains just a set on which G acts 2-transitively. In our case, we also have that the 2-point stabilizer is infinite and abelian. Organizers: Tuna Altınel (Lyon), Ayşe Berkman (İstanbul), Özlem Beyarslan (İstanbul),

BN-Pair 19-22 October 2016, Istanbul

Conference in honour of 60th birthdays of Alexandre Borovik and Ali Nesin 19-22 October 2016, Istanbul

Organizers: Tuna Altınel (Lyon), Ayşe Berkman (İstanbul), Özlem Beyarslan (İstanbul),

of Tits rank 1, G is PSL2 (K). Done.

(iii) By the theory of split Organizers: Tuna Altınel (Lyon), Ayşe Berkman (İstanbul), Özlem Beyarslan (İstanbul),

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo Summary Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Summary

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo Summary Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Summary CiBo1 :

Trace of geometry (of involutions or strongly real elements): BN-pais of rank 1

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo Summary Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Summary CiBo1 :

Trace of geometry (of involutions or strongly real elements): BN-pais of rank 1

CiBo2 :

PSL2 -like rank computation

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo Summary Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Summary CiBo1 :

Trace of geometry (of involutions or strongly real elements): BN-pais of rank 1

CiBo2 :

PSL2 -like rank computation (but there is definitely some geometry here too. . . )

Joshua Wiscons

Geometries and small groups of fMr

Groups of Morley rank 5 - CiBo Summary Theorem (Deloro-Wiscons ‘16) A simple group of Morley rank 5 is bad. Summary CiBo1 :

Trace of geometry (of involutions or strongly real elements): BN-pais of rank 1

CiBo2 :

PSL2 -like rank computation (but there is definitely some geometry here too. . . )

CiBo3 :

Geometry of involutions: C-lines, B-lines, and Bachmann

Joshua Wiscons

Geometries and small groups of fMr

Act IV Applications to N-groups - 2 CiBo2

Joshua Wiscons

Geometries and small groups of fMr

Adrien’s Observation Setting (CiBo2 ) G is an N-group of odd type for which C◦ (i) is a Borel inverted by any ω ∈ C(i) − {i}, and S = Z2∞ o hωi

Joshua Wiscons

Geometries and small groups of fMr

Adrien’s Observation Setting (CiBo2 ) G is an N-group of odd type for which C◦ (i) is a Borel inverted by any ω ∈ C(i) − {i}, and S = Z2∞ o hωi Observe that. . .

Joshua Wiscons

Geometries and small groups of fMr

Adrien’s Observation Setting (CiBo2 ) G is an N-group of odd type for which C◦ (i) is a Borel inverted by any ω ∈ C(i) − {i}, and S = Z2∞ o hωi Observe that. . . 1

By general principles, if g ∈ G is generic, then there is exactly one k ∈ I(G) such that g ∈ C◦ (k).

Joshua Wiscons

Geometries and small groups of fMr

Adrien’s Observation Setting (CiBo2 ) G is an N-group of odd type for which C◦ (i) is a Borel inverted by any ω ∈ C(i) − {i}, and S = Z2∞ o hωi Observe that. . . 1

2

By general principles, if g ∈ G is generic, then there is exactly one k ∈ I(G) such that g ∈ C◦ (k). Since C◦ (i) is inverted by any ω ∈ C(i) − {i}, the generic element of G is a product of involutions.

Joshua Wiscons

Geometries and small groups of fMr

Adrien’s Observation Setting (CiBo2 ) G is an N-group of odd type for which C◦ (i) is a Borel inverted by any ω ∈ C(i) − {i}, and S = Z2∞ o hωi Observe that. . . 1

2

By general principles, if g ∈ G is generic, then there is exactly one k ∈ I(G) such that g ∈ C◦ (k). Since C◦ (i) is inverted by any ω ∈ C(i) − {i}, the generic element of G is a product of involutions.

Let i, j be a generic pair of involutions.

Joshua Wiscons

Geometries and small groups of fMr

Adrien’s Observation Setting (CiBo2 ) G is an N-group of odd type for which C◦ (i) is a Borel inverted by any ω ∈ C(i) − {i}, and S = Z2∞ o hωi Observe that. . . 1

2

By general principles, if g ∈ G is generic, then there is exactly one k ∈ I(G) such that g ∈ C◦ (k). Since C◦ (i) is inverted by any ω ∈ C(i) − {i}, the generic element of G is a product of involutions.

Let i, j be a generic pair of involutions. There is exactly one involution k such that ij ∈ C◦ (k).

Joshua Wiscons

Geometries and small groups of fMr

Adrien’s Observation Setting (CiBo2 ) G is an N-group of odd type for which C◦ (i) is a Borel inverted by any ω ∈ C(i) − {i}, and S = Z2∞ o hωi Observe that. . . 1

2

By general principles, if g ∈ G is generic, then there is exactly one k ∈ I(G) such that g ∈ C◦ (k). Since C◦ (i) is inverted by any ω ∈ C(i) − {i}, the generic element of G is a product of involutions.

Let i, j be a generic pair of involutions. There is exactly one involution k such that ij ∈ C◦ (k). Note: ij ∈ C◦ (k) iff both i and j invert C◦ (k) (using CiBo2 )

Joshua Wiscons

Geometries and small groups of fMr

Adrien’s Observation Setting (CiBo2 ) G is an N-group of odd type for which C◦ (i) is a Borel inverted by any ω ∈ C(i) − {i}, and S = Z2∞ o hωi Observe that. . . 1

2

By general principles, if g ∈ G is generic, then there is exactly one k ∈ I(G) such that g ∈ C◦ (k). Since C◦ (i) is inverted by any ω ∈ C(i) − {i}, the generic element of G is a product of involutions.

Let i, j be a generic pair of involutions. There is exactly one involution k such that ij ∈ C◦ (k). Note: ij ∈ C◦ (k) iff both i and j invert C◦ (k) (using CiBo2 ) So, ij ∈ C◦ (k) iff both i and j centralize k Joshua Wiscons

Geometries and small groups of fMr

Adrien’s Observation Setting (CiBo2 ) G is an N-group of odd type for which C◦ (i) is a Borel inverted by any ω ∈ C(i) − {i}, and S = Z2∞ o hωi

Let i, j be a generic pair of involutions. There is exactly one involution k such that ij ∈ C◦ (k). Note: ij ∈ C◦ (k) iff both i and j invert C◦ (k) (using CiBo2 ) So, ij ∈ C◦ (k) iff both i and j centralize k Joshua Wiscons

Geometries and small groups of fMr

Adrien’s Observation Setting (CiBo2 ) G is an N-group of odd type for which C◦ (i) is a Borel inverted by any ω ∈ C(i) − {i}, and S = Z2∞ o hωi Let i, j be a generic pair of involutions. There is exactly one involution k such that ij ∈ C◦ (k). Note: ij ∈ C◦ (k) iff both i and j invert C◦ (k) (using CiBo2 ) So, ij ∈ C◦ (k) iff both i and j centralize k

Joshua Wiscons

Geometries and small groups of fMr

Adrien’s Observation Setting (CiBo2 ) G is an N-group of odd type for which C◦ (i) is a Borel inverted by any ω ∈ C(i) − {i}, and S = Z2∞ o hωi Let i, j be a generic pair of involutions. There is exactly one involution k such that ij ∈ C◦ (k). Note: ij ∈ C◦ (k) iff both i and j invert C◦ (k) (using CiBo2 ) So, ij ∈ C◦ (k) iff both i and j centralize k

Lemma (Adrien’s Observation)

Joshua Wiscons

Geometries and small groups of fMr

Adrien’s Observation Setting (CiBo2 ) G is an N-group of odd type for which C◦ (i) is a Borel inverted by any ω ∈ C(i) − {i}, and S = Z2∞ o hωi Let i, j be a generic pair of involutions. There is exactly one involution k such that ij ∈ C◦ (k). Note: ij ∈ C◦ (k) iff both i and j invert C◦ (k) (using CiBo2 ) So, ij ∈ C◦ (k) iff both i and j centralize k

Lemma (Adrien’s Observation) If i, j is a generic pair of involutions, then C(i) − {i} and C(j) − {j} contain a unique common involution.

Joshua Wiscons

Geometries and small groups of fMr

Adrien’s Observation Setting (CiBo2 ) G is an N-group of odd type for which C◦ (i) is a Borel inverted by any ω ∈ C(i) − {i}, and S = Z2∞ o hωi Let i, j be a generic pair of involutions. There is exactly one involution k such that ij ∈ C◦ (k). Note: ij ∈ C◦ (k) iff both i and j invert C◦ (k) (using CiBo2 ) So, ij ∈ C◦ (k) iff both i and j centralize k

Lemma (Adrien’s Observation) If i, j is a generic pair of involutions, then C(i) − {i} and C(j) − {j} contain a unique common involution. That is,

Joshua Wiscons

Geometries and small groups of fMr

Adrien’s Observation Setting (CiBo2 ) G is an N-group of odd type for which C◦ (i) is a Borel inverted by any ω ∈ C(i) − {i}, and S = Z2∞ o hωi Let i, j be a generic pair of involutions. There is exactly one involution k such that ij ∈ C◦ (k). Note: ij ∈ C◦ (k) iff both i and j invert C◦ (k) (using CiBo2 ) So, ij ∈ C◦ (k) iff both i and j centralize k

Lemma (Adrien’s Observation) If i, j is a generic pair of involutions, then C(i) − {i} and C(j) − {j} contain a unique common involution. That is, a generic pair of C-lines intersect in a unique involution, and hence, Joshua Wiscons

Geometries and small groups of fMr

Adrien’s Observation Setting (CiBo2 ) G is an N-group of odd type for which C◦ (i) is a Borel inverted by any ω ∈ C(i) − {i}, and S = Z2∞ o hωi Let i, j be a generic pair of involutions. There is exactly one involution k such that ij ∈ C◦ (k). Note: ij ∈ C◦ (k) iff both i and j invert C◦ (k) (using CiBo2 ) So, ij ∈ C◦ (k) iff both i and j centralize k

Lemma (Adrien’s Observation) If i, j is a generic pair of involutions, then C(i) − {i} and C(j) − {j} contain a unique common involution. That is, a generic pair of C-lines intersect in a unique involution, and hence, a generic pair of involutions define a unique C-line. Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry Lemma (Adrien’s Observation) If G is a CiBo2 group, then the geometry of involutions with respect to C-lines is that of a generically defined projective plane.

Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry Lemma (Adrien’s Observation) If G is a CiBo2 group, then the geometry of involutions with respect to C-lines is that of a generically defined projective plane.

Recall “Bachmann’s Theorem”:

Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry Lemma (Adrien’s Observation) If G is a CiBo2 group, then the geometry of involutions with respect to C-lines is that of a generically defined projective plane.

Recall “Bachmann’s Theorem”: Corollary (“Bachmann’s Theorem”) For a group of fMr, the geometry of involutions with respect to B-collinearity can not be that of a projective plane.

Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry Lemma (Adrien’s Observation) If G is a CiBo2 group, then the geometry of involutions with respect to C-lines is that of a generically defined projective plane.

Recall “Bachmann’s Theorem”: Corollary (“Bachmann’s Theorem”) For a group of fMr, the geometry of involutions with respect to B-collinearity can not be that of a projective plane. Also, recall that the proof was roughly as follows:

Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry Lemma (Adrien’s Observation) If G is a CiBo2 group, then the geometry of involutions with respect to C-lines is that of a generically defined projective plane.

Recall “Bachmann’s Theorem”: Corollary (“Bachmann’s Theorem”) For a group of fMr, the geometry of involutions with respect to B-collinearity can not be that of a projective plane. Also, recall that the proof was roughly as follows: 1 Show that B-collinearity is equivalent to C-collinearity

Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry Lemma (Adrien’s Observation) If G is a CiBo2 group, then the geometry of involutions with respect to C-lines is that of a generically defined projective plane.

Recall “Bachmann’s Theorem”: Corollary (“Bachmann’s Theorem”) For a group of fMr, the geometry of involutions with respect to B-collinearity can not be that of a projective plane. Also, recall that the proof was roughly as follows: 1 Show that B-collinearity is equivalent to C-collinearity 2 Embed the plane in a 3-dimensional (Desarguesian) space where Points: elements of G Lines: G-translates, `g, of lines from the plane associated to I Planes: G-translates, Ig, of the plane I

Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry Lemma (Adrien’s Observation) If G is a CiBo2 group, then the geometry of involutions with respect to C-lines is that of a generically defined projective plane.

Recall “Bachmann’s Theorem”: Corollary (“Bachmann’s Theorem”) For a group of fMr, the geometry of involutions with respect to B-collinearity can not be that of a projective plane. Also, recall that the proof was roughly as follows: 1 Show that B-collinearity is equivalent to C-collinearity 2 Embed the plane in a 3-dimensional (Desarguesian) space where Points: elements of G Lines: G-translates, `g, of lines from the plane associated to I Planes: G-translates, Ig, of the plane I 3

Exploit the fact that G acts regularly on the points of the geometry Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry Lemma (Adrien’s Observation) If G is a CiBo2 group, then the geometry of involutions with respect to C-lines is that of a generically defined projective plane.

Recall “Bachmann’s Theorem”: Corollary (“Bachmann’s Theorem”) For a group of fMr, the geometry of involutions with respect to B-collinearity can not be that of a projective plane. Also, recall that the proof was roughly as follows: Show that B-collinearity is equivalent to C-collinearity 2 Embed the plane in a 3-dimensional (Desarguesian) space where

X1

Points: elements of G Lines: G-translates, `g, of lines from the plane associated to I Planes: G-translates, Ig, of the plane I 3

Exploit the fact that G acts regularly on the points of the geometry Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry Lemma (Adrien’s Observation) If G is a CiBo2 group, then the geometry of involutions with respect to C-lines is that of a generically defined projective plane.

Recall “Bachmann’s Theorem”: Corollary (“Bachmann’s Theorem”) For a group of fMr, the geometry of involutions with respect to B-collinearity can not be that of a projective plane. Also, recall that the proof was roughly as follows: Show that B-collinearity is equivalent to C-collinearity → 2 Embed the plane in a 3-dimensional (Desarguesian) space where

X1

Points: elements of G Lines: G-translates, `g, of lines from the plane associated to I Planes: G-translates, Ig, of the plane I 3

Exploit the fact that G acts regularly on the points of the geometry Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry Lemma (Adrien’s Observation) If G is a CiBo2 group, then the geometry of involutions with respect to C-lines is that of a generically defined projective plane.

Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry Lemma (Adrien’s Observation)

Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry Lemma (Adrien’s Observation) Let G be a CiBo2 group, and define points, lines and planes as follows. Points - P: elements of G Lines - L: G-translates, `g, of lines from the plane associated to I Planes - Π: G-translates, Ig, of the plane I

Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry Lemma (Adrien’s Observation) Let G be a CiBo2 group, and define points, lines and planes as follows. Points - P: elements of G Lines - L: G-translates, `g, of lines from the plane associated to I Planes - Π: G-translates, Ig, of the plane I

Then the geometry is that of a generically projective 3-space. Specifically,

Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry Lemma (Adrien’s Observation) Let G be a CiBo2 group, and define points, lines and planes as follows. Points - P: elements of G Lines - L: G-translates, `g, of lines from the plane associated to I Planes - Π: G-translates, Ig, of the plane I

Then the geometry is that of a generically projective 3-space. Specifically, 0

the sets P, L, and Π each have degree 1,

Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry Lemma (Adrien’s Observation) Let G be a CiBo2 group, and define points, lines and planes as follows. Points - P: elements of G Lines - L: G-translates, `g, of lines from the plane associated to I Planes - Π: G-translates, Ig, of the plane I

Then the geometry is that of a generically projective 3-space. Specifically, 0

the sets P, L, and Π each have degree 1,

1

generically points are uniquely collinear,

Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry Lemma (Adrien’s Observation) Let G be a CiBo2 group, and define points, lines and planes as follows. Points - P: elements of G Lines - L: G-translates, `g, of lines from the plane associated to I Planes - Π: G-translates, Ig, of the plane I

Then the geometry is that of a generically projective 3-space. Specifically, 0

the sets P, L, and Π each have degree 1,

1

generically points are uniquely collinear,

2

generically a point and a line are uniquely coplanar,

Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry Lemma (Adrien’s Observation) Let G be a CiBo2 group, and define points, lines and planes as follows. Points - P: elements of G Lines - L: G-translates, `g, of lines from the plane associated to I Planes - Π: G-translates, Ig, of the plane I

Then the geometry is that of a generically projective 3-space. Specifically, 0

the sets P, L, and Π each have degree 1,

1

generically points are uniquely collinear,

2

generically a point and a line are uniquely coplanar,

3

generically a line intersects a plane,

Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry Lemma (Adrien’s Observation) Let G be a CiBo2 group, and define points, lines and planes as follows. Points - P: elements of G Lines - L: G-translates, `g, of lines from the plane associated to I Planes - Π: G-translates, Ig, of the plane I

Then the geometry is that of a generically projective 3-space. Specifically, 0

the sets P, L, and Π each have degree 1,

1

generically points are uniquely collinear,

2

generically a point and a line are uniquely coplanar,

3

generically a line intersects a plane,

4

generically two planes intersect in at least a line,

Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry Lemma (Adrien’s Observation) Let G be a CiBo2 group, and define points, lines and planes as follows. Points - P: elements of G Lines - L: G-translates, `g, of lines from the plane associated to I Planes - Π: G-translates, Ig, of the plane I

Then the geometry is that of a generically projective 3-space. Specifically, 0

the sets P, L, and Π each have degree 1,

1

generically points are uniquely collinear,

2

generically a point and a line are uniquely coplanar,

3

generically a line intersects a plane,

4

generically two planes intersect in at least a line,

5

generically four points form a non-coplanar quadrilateral, and

Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry Lemma (Adrien’s Observation) Let G be a CiBo2 group, and define points, lines and planes as follows. Points - P: elements of G Lines - L: G-translates, `g, of lines from the plane associated to I Planes - Π: G-translates, Ig, of the plane I

Then the geometry is that of a generically projective 3-space. Specifically, 0

the sets P, L, and Π each have degree 1,

1

generically points are uniquely collinear,

2

generically a point and a line are uniquely coplanar,

3

generically a line intersects a plane,

4

generically two planes intersect in at least a line,

5

generically four points form a non-coplanar quadrilateral, and

6

generically a line contains at least 3 points. Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry So what now?

Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry So what now? Option 1: The Boris Weisfeiler approach: fill in the missing intersections.

Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry So what now? Option 1: The Boris Weisfeiler approach: fill in the missing intersections. In the (finite) case of PGL2 (q) with q odd, the geometry of involutions is also that of a “generically defined” projective plane.

Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry So what now? Option 1: The Boris Weisfeiler approach: fill in the missing intersections. In the (finite) case of PGL2 (q) with q odd, the geometry of involutions is also that of a “generically defined” projective plane. The missing points are unipotent subgroups.

Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry So what now? Option 1: The Boris Weisfeiler approach: fill in the missing intersections. In the (finite) case of PGL2 (q) with q odd, the geometry of involutions is also that of a “generically defined” projective plane. The missing points are unipotent subgroups. This is all related to the following generalization of Bachmann.

Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry So what now? Option 1: The Boris Weisfeiler approach: fill in the missing intersections. In the (finite) case of PGL2 (q) with q odd, the geometry of involutions is also that of a “generically defined” projective plane. The missing points are unipotent subgroups. This is all related to the following generalization of Bachmann.

Theorem (Schröder ’82) Let Γ be a projective plane, and let Ω be a set of points that contains at most two points of any line. Assume further that the points P − Ω are in a one-to-one correspondence with the set I of involutions of some group G where collinearity is B-collinearity. Then there exist a field K and a quadratic form Q on the 3-dimensional vector space K 3 such that Γ = P2 (K) and Ω is the quadric given by Q(x) = 0.

Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry So what now? Option 1: The Boris Weisfeiler approach: fill in the missing intersections.

Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry So what now? Option 1: The Boris Weisfeiler approach: fill in the missing intersections. Option 2: Work generically and try to coordinatize the 3-space.

Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry So what now? Option 1: The Boris Weisfeiler approach: fill in the missing intersections. Option 2: Work generically and try to coordinatize the 3-space. Need an appropriate version of Desargues, which is probably not the following.

Joshua Wiscons

Geometries and small groups of fMr

CiBo2 Geometry So what now? Option 1: The Boris Weisfeiler approach: fill in the missing intersections. Option 2: Work generically and try to coordinatize the 3-space. Need an appropriate version of Desargues, which is probably not the following. Lemma (Desargues’ Theorem) Suppose that the geometry of CiBo2 minimal simple group contains the following Desarguesian configuration lying in a plane, and assume that at least two of C1 , C2 , and C3 are collinear. Then all three are collinear. C2 A1 O

B1

C1 B2

A2 A3

Joshua Wiscons

B3

C3

Geometries and small groups of fMr

CiBo2 Geometry So what now? Option 1: The Boris Weisfeiler approach: fill in the missing intersections. Option 2: Work generically and try to coordinatize the 3-space. Need an appropriate version of Desargues, which is probably not the following. Lemma (Desargues’ Theorem) Suppose that the geometry of CiBo2 minimal simple group contains the following Desarguesian configuration lying in a plane, and assume that at least two of C1 , C2 , and C3 are collinear. Then all three are collinear. C2 A1 O

B1

C1 B2

A2 A3

B3

C3

Keep going, e.g. try to define translations. Joshua Wiscons

Geometries and small groups of fMr

Final thoughts Thought The geometry of involutions is highly relevant to the study of “small” groups, and deserves to be better understood especially in the generically defined case.

Joshua Wiscons

Geometries and small groups of fMr

Final thoughts Thought The geometry of involutions is highly relevant to the study of “small” groups, and deserves to be better understood especially in the generically defined case. As (a tiny bit of) evidence, we saw that

Joshua Wiscons

Geometries and small groups of fMr

Final thoughts Thought The geometry of involutions is highly relevant to the study of “small” groups, and deserves to be better understood especially in the generically defined case. As (a tiny bit of) evidence, we saw that CiBo3 has been successfully treated in rank 5 with these geometries

Joshua Wiscons

Geometries and small groups of fMr

Final thoughts Thought The geometry of involutions is highly relevant to the study of “small” groups, and deserves to be better understood especially in the generically defined case. As (a tiny bit of) evidence, we saw that CiBo3 has been successfully treated in rank 5 with these geometries CiBo2 is clearly tied up with (generically defined versions) of them

Joshua Wiscons

Geometries and small groups of fMr

Final thoughts Thought The geometry of involutions is highly relevant to the study of “small” groups, and deserves to be better understood especially in the generically defined case. As (a tiny bit of) evidence, we saw that CiBo3 has been successfully treated in rank 5 with these geometries CiBo2 is clearly tied up with (generically defined versions) of them CiBo1 was treated with the theory of split BN-pairs of rank 1, but. . .

Joshua Wiscons

Geometries and small groups of fMr

Final thoughts Thought The geometry of involutions is highly relevant to the study of “small” groups, and deserves to be better understood especially in the generically defined case. As (a tiny bit of) evidence, we saw that CiBo3 has been successfully treated in rank 5 with these geometries CiBo2 is clearly tied up with (generically defined versions) of them CiBo1 was treated with the theory of split BN-pairs of rank 1, but. . . Nesin analyzed sharply 2-transitive groups via the geometry of B-lines

Joshua Wiscons

Geometries and small groups of fMr

Final thoughts Thought The geometry of involutions is highly relevant to the study of “small” groups, and deserves to be better understood especially in the generically defined case. As (a tiny bit of) evidence, we saw that CiBo3 has been successfully treated in rank 5 with these geometries CiBo2 is clearly tied up with (generically defined versions) of them CiBo1 was treated with the theory of split BN-pairs of rank 1, but. . . Nesin analyzed sharply 2-transitive groups via the geometry of B-lines Sharply 2-transitive groups are also split BN-pairs of rank 1 (and the most degenerate of all BN-pairs)

Joshua Wiscons

Geometries and small groups of fMr

Final thoughts Thought The geometry of involutions is highly relevant to the study of “small” groups, and deserves to be better understood especially in the generically defined case. As (a tiny bit of) evidence, we saw that CiBo3 has been successfully treated in rank 5 with these geometries CiBo2 is clearly tied up with (generically defined versions) of them CiBo1 was treated with the theory of split BN-pairs of rank 1, but. . . Nesin analyzed sharply 2-transitive groups via the geometry of B-lines Sharply 2-transitive groups are also split BN-pairs of rank 1 (and the most degenerate of all BN-pairs) This approach was also used to study simple groups of Morley rank 4

Joshua Wiscons

Geometries and small groups of fMr

Final thoughts

Thought (Hopes and Dreams and Ignorance) So, perhaps there is hope to make a little progress on some long-standing bad configurations in groups of fMr.

Joshua Wiscons

Geometries and small groups of fMr

Thank You

Joshua Wiscons

Geometries and small groups of fMr