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Classifying space for proper actions for groups admitting a strict fundamental domain Tomasz Prytu la 04.04.2018 joint work with Nansen Petrosyan joint work with Nansen Petrosyan Outline joint work with Nansen Petrosyan Outline 1

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Classifying space for proper actions for groups admitting a strict fundamental domain

Tomasz Prytu la 04.04.2018

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joint work with Nansen Petrosyan

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joint work with Nansen Petrosyan

Outline

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joint work with Nansen Petrosyan

Outline

1 Classifying space for proper actions EG

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joint work with Nansen Petrosyan

Outline

1 Classifying space for proper actions EG 2 Davis complex for a Coxeter group

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joint work with Nansen Petrosyan

Outline

1 Classifying space for proper actions EG 2 Davis complex for a Coxeter group 3 Our construction

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joint work with Nansen Petrosyan

Outline

1 Classifying space for proper actions EG 2 Davis complex for a Coxeter group 3 Our construction 4 Applications

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Classifying space for proper actions

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Classifying space for proper actions

Let G be an infinite discrete group.

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Classifying space for proper actions

Let G be an infinite discrete group.

Definition

A model for the classifying space EG is a G–CW–complex X such that:

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Classifying space for proper actions

Let G be an infinite discrete group.

Definition

A model for the classifying space EG is a G–CW–complex X such that:

◮ G X is proper (finite cell stabilisers)

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Classifying space for proper actions

Let G be an infinite discrete group.

Definition

A model for the classifying space EG is a G–CW–complex X such that:

◮ G X is proper (finite cell stabilisers) ◮ for every finite subgroup F ⊂ G the fixed point set X F is

contractible (= ∅)

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Classifying space for proper actions

Let G be an infinite discrete group.

Definition

A model for the classifying space EG is a G–CW–complex X such that:

◮ G X is proper (finite cell stabilisers) ◮ for every finite subgroup F ⊂ G the fixed point set X F is

contractible (= ∅)

Remark

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Classifying space for proper actions

Let G be an infinite discrete group.

Definition

A model for the classifying space EG is a G–CW–complex X such that:

◮ G X is proper (finite cell stabilisers) ◮ for every finite subgroup F ⊂ G the fixed point set X F is

contractible (= ∅)

Remark

◮ EG always exists

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Classifying space for proper actions

Let G be an infinite discrete group.

Definition

A model for the classifying space EG is a G–CW–complex X such that:

◮ G X is proper (finite cell stabilisers) ◮ for every finite subgroup F ⊂ G the fixed point set X F is

contractible (= ∅)

Remark

◮ EG always exists ◮ any two models for EG are G–homotopy equivalent

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Classifying space for proper actions

Examples

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Classifying space for proper actions

Examples

1. R

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Classifying space for proper actions

Examples

1. R s

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Classifying space for proper actions

Examples

1. R s t

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Classifying space for proper actions

Examples

1. R s t

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Classifying space for proper actions

Examples

1. R s t D∞ = s, t ⊂ Isom(R)

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Classifying space for proper actions

Examples

1. R s t D∞ = s, t ⊂ Isom(R) D∞ R properly,

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Classifying space for proper actions

Examples

1. R s t D∞ = s, t ⊂ Isom(R) D∞ R properly, Rs ≈ ∗,

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Classifying space for proper actions

Examples

1. R s t D∞ = s, t ⊂ Isom(R) D∞ R properly, Rs ≈ ∗, R ≃ ED∞

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Classifying space for proper actions

Examples

1. R s t D∞ = s, t ⊂ Isom(R) D∞ R properly, Rs ≈ ∗, R ≃ ED∞

  • 2. Let G act properly on a tree T.
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Classifying space for proper actions

Examples

1. R s t D∞ = s, t ⊂ Isom(R) D∞ R properly, Rs ≈ ∗, R ≃ ED∞

  • 2. Let G act properly on a tree T. Then T ≃ EG.
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Classifying space for proper actions

Examples

1. R s t D∞ = s, t ⊂ Isom(R) D∞ R properly, Rs ≈ ∗, R ≃ ED∞

  • 2. Let G act properly on a tree T. Then T ≃ EG.

◮ every finite F ⊂ G has a fixed point (Fixed Point Theorem)

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Classifying space for proper actions

Examples

1. R s t D∞ = s, t ⊂ Isom(R) D∞ R properly, Rs ≈ ∗, R ≃ ED∞

  • 2. Let G act properly on a tree T. Then T ≃ EG.

◮ every finite F ⊂ G has a fixed point (Fixed Point Theorem) ◮ every fixed set is contractible.

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Classifying space for proper actions

Examples

1. R s t D∞ = s, t ⊂ Isom(R) D∞ R properly, Rs ≈ ∗, R ≃ ED∞

  • 2. Let G act properly on a tree T. Then T ≃ EG.

◮ every finite F ⊂ G has a fixed point (Fixed Point Theorem) ◮ every fixed set is contractible.

  • 3. G X properly, X - CAT(0) space.
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Classifying space for proper actions

Examples

1. R s t D∞ = s, t ⊂ Isom(R) D∞ R properly, Rs ≈ ∗, R ≃ ED∞

  • 2. Let G act properly on a tree T. Then T ≃ EG.

◮ every finite F ⊂ G has a fixed point (Fixed Point Theorem) ◮ every fixed set is contractible.

  • 3. G X properly, X - CAT(0) space. Then X ≃ EG.
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Classifying space for proper actions

Examples

1. R s t D∞ = s, t ⊂ Isom(R) D∞ R properly, Rs ≈ ∗, R ≃ ED∞

  • 2. Let G act properly on a tree T. Then T ≃ EG.

◮ every finite F ⊂ G has a fixed point (Fixed Point Theorem) ◮ every fixed set is contractible.

  • 3. G X properly, X - CAT(0) space. Then X ≃ EG.
  • 4. G is δ–hyperbolic.
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Classifying space for proper actions

Examples

1. R s t D∞ = s, t ⊂ Isom(R) D∞ R properly, Rs ≈ ∗, R ≃ ED∞

  • 2. Let G act properly on a tree T. Then T ≃ EG.

◮ every finite F ⊂ G has a fixed point (Fixed Point Theorem) ◮ every fixed set is contractible.

  • 3. G X properly, X - CAT(0) space. Then X ≃ EG.
  • 4. G is δ–hyperbolic. Then the Rips complex Pr(G) ≃ EG.
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Classifying space for proper actions

Examples

1. R s t D∞ = s, t ⊂ Isom(R) D∞ R properly, Rs ≈ ∗, R ≃ ED∞

  • 2. Let G act properly on a tree T. Then T ≃ EG.

◮ every finite F ⊂ G has a fixed point (Fixed Point Theorem) ◮ every fixed set is contractible.

  • 3. G X properly, X - CAT(0) space. Then X ≃ EG.
  • 4. G is δ–hyperbolic. Then the Rips complex Pr(G) ≃ EG.
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Classifying space for proper actions

Goal

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Classifying space for proper actions

Goal

Construct possibly simple models for EG

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Classifying space for proper actions

Goal

Construct possibly simple models for EG of small dimension =

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Classifying space for proper actions

Goal

Construct possibly simple models for EG of small dimension = vcdG - virtual cohomological dimension

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Classifying space for proper actions

Goal

Construct possibly simple models for EG of small dimension = vcdG - virtual cohomological dimension cdG - Bredon cohomological dimension

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Classifying space for proper actions

Goal

Construct possibly simple models for EG of small dimension = vcdG - virtual cohomological dimension cdG - Bredon cohomological dimension

Homology HG

∗ (EG)

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Classifying space for proper actions

Goal

Construct possibly simple models for EG of small dimension = vcdG - virtual cohomological dimension cdG - Bredon cohomological dimension

Homology HG

∗ (EG) ◮ Baum-Connes conjecture

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Classifying space for proper actions

Goal

Construct possibly simple models for EG of small dimension = vcdG - virtual cohomological dimension cdG - Bredon cohomological dimension

Homology HG

∗ (EG) ◮ Baum-Connes conjecture

K G

∗ (EG) ∼ =

− → K∗(C ∗

r (G))

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Classifying space for proper actions

Goal

Construct possibly simple models for EG of small dimension = vcdG - virtual cohomological dimension cdG - Bredon cohomological dimension

Homology HG

∗ (EG) ◮ Baum-Connes conjecture

K G

∗ (EG) ∼ =

− → K∗(C ∗

r (G)) ◮ Atiyah-Segal Completion theorem (L¨

uck-Oliver ’01)

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Classifying space for proper actions

Goal

Construct possibly simple models for EG of small dimension = vcdG - virtual cohomological dimension cdG - Bredon cohomological dimension

Homology HG

∗ (EG) ◮ Baum-Connes conjecture

K G

∗ (EG) ∼ =

− → K∗(C ∗

r (G)) ◮ Atiyah-Segal Completion theorem (L¨

uck-Oliver ’01)

Example

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Classifying space for proper actions

Goal

Construct possibly simple models for EG of small dimension = vcdG - virtual cohomological dimension cdG - Bredon cohomological dimension

Homology HG

∗ (EG) ◮ Baum-Connes conjecture

K G

∗ (EG) ∼ =

− → K∗(C ∗

r (G)) ◮ Atiyah-Segal Completion theorem (L¨

uck-Oliver ’01)

Example

SL(2, Z)

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Classifying space for proper actions

Goal

Construct possibly simple models for EG of small dimension = vcdG - virtual cohomological dimension cdG - Bredon cohomological dimension

Homology HG

∗ (EG) ◮ Baum-Connes conjecture

K G

∗ (EG) ∼ =

− → K∗(C ∗

r (G)) ◮ Atiyah-Segal Completion theorem (L¨

uck-Oliver ’01)

Example

SL(2, Z) properly by isometries on H2

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Classifying space for proper actions

Goal

Construct possibly simple models for EG of small dimension = vcdG - virtual cohomological dimension cdG - Bredon cohomological dimension

Homology HG

∗ (EG) ◮ Baum-Connes conjecture

K G

∗ (EG) ∼ =

− → K∗(C ∗

r (G)) ◮ Atiyah-Segal Completion theorem (L¨

uck-Oliver ’01)

Example

SL(2, Z) properly by isometries on H2 ⇒ H2 ≃ ESL(2, Z)

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Classifying space for proper actions

Goal

Construct possibly simple models for EG of small dimension = vcdG - virtual cohomological dimension cdG - Bredon cohomological dimension

Homology HG

∗ (EG) ◮ Baum-Connes conjecture

K G

∗ (EG) ∼ =

− → K∗(C ∗

r (G)) ◮ Atiyah-Segal Completion theorem (L¨

uck-Oliver ’01)

Example

SL(2, Z) properly by isometries on H2 ⇒ H2 ≃ ESL(2, Z) SL(2, Z) ∼ = Z/4 ∗Z/2 Z/6

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Classifying space for proper actions

Goal

Construct possibly simple models for EG of small dimension = vcdG - virtual cohomological dimension cdG - Bredon cohomological dimension

Homology HG

∗ (EG) ◮ Baum-Connes conjecture

K G

∗ (EG) ∼ =

− → K∗(C ∗

r (G)) ◮ Atiyah-Segal Completion theorem (L¨

uck-Oliver ’01)

Example

SL(2, Z) properly by isometries on H2 ⇒ H2 ≃ ESL(2, Z) SL(2, Z) ∼ = Z/4 ∗Z/2 Z/6 properly on a tree

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Classifying space for proper actions

Goal

Construct possibly simple models for EG of small dimension = vcdG - virtual cohomological dimension cdG - Bredon cohomological dimension

Homology HG

∗ (EG) ◮ Baum-Connes conjecture

K G

∗ (EG) ∼ =

− → K∗(C ∗

r (G)) ◮ Atiyah-Segal Completion theorem (L¨

uck-Oliver ’01)

Example

SL(2, Z) properly by isometries on H2 ⇒ H2 ≃ ESL(2, Z) SL(2, Z) ∼ = Z/4 ∗Z/2 Z/6 properly on a tree ⇒ Tree ≃ ESL(2, Z)

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Davis complex for a Coxeter group

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Davis complex for a Coxeter group

Right-Angled Coxeter groups

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Davis complex for a Coxeter group

Right-Angled Coxeter groups

Let L be a finite, flag simplicial complex.

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Davis complex for a Coxeter group

Right-Angled Coxeter groups

Let L be a finite, flag simplicial complex.

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Davis complex for a Coxeter group

Right-Angled Coxeter groups

Let L be a finite, flag simplicial complex.

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Davis complex for a Coxeter group

Right-Angled Coxeter groups

Let L be a finite, flag simplicial complex. W = WL = si ∈ V (L) | s2

i = e, sisj = sjsi iff {si, sj} ∈ E(L)

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Davis complex for a Coxeter group

Right-Angled Coxeter groups

Let L be a finite, flag simplicial complex. W = WL = si ∈ V (L) | s2

i = e, sisj = sjsi iff {si, sj} ∈ E(L)

Examples

L WL

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Davis complex for a Coxeter group

Right-Angled Coxeter groups

Let L be a finite, flag simplicial complex. W = WL = si ∈ V (L) | s2

i = e, sisj = sjsi iff {si, sj} ∈ E(L)

Examples

L ∆n WL

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Davis complex for a Coxeter group

Right-Angled Coxeter groups

Let L be a finite, flag simplicial complex. W = WL = si ∈ V (L) | s2

i = e, sisj = sjsi iff {si, sj} ∈ E(L)

Examples

L ∆n WL (Z/2)n+1

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Davis complex for a Coxeter group

Right-Angled Coxeter groups

Let L be a finite, flag simplicial complex. W = WL = si ∈ V (L) | s2

i = e, sisj = sjsi iff {si, sj} ∈ E(L)

Examples

L ∆n (∆n)(0) WL (Z/2)n+1

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Davis complex for a Coxeter group

Right-Angled Coxeter groups

Let L be a finite, flag simplicial complex. W = WL = si ∈ V (L) | s2

i = e, sisj = sjsi iff {si, sj} ∈ E(L)

Examples

L ∆n (∆n)(0) WL (Z/2)n+1 (Z/2)∗(n+1)

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Davis complex for a Coxeter group

Right-Angled Coxeter groups

Let L be a finite, flag simplicial complex. W = WL = si ∈ V (L) | s2

i = e, sisj = sjsi iff {si, sj} ∈ E(L)

Examples

L ∆n (∆n)(0) L1 ⊔ L2 WL (Z/2)n+1 (Z/2)∗(n+1)

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Davis complex for a Coxeter group

Right-Angled Coxeter groups

Let L be a finite, flag simplicial complex. W = WL = si ∈ V (L) | s2

i = e, sisj = sjsi iff {si, sj} ∈ E(L)

Examples

L ∆n (∆n)(0) L1 ⊔ L2 WL (Z/2)n+1 (Z/2)∗(n+1) WL1 ∗ WL2

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Davis complex for a Coxeter group

Right-Angled Coxeter groups

Let L be a finite, flag simplicial complex. W = WL = si ∈ V (L) | s2

i = e, sisj = sjsi iff {si, sj} ∈ E(L)

Examples

L ∆n (∆n)(0) L1 ⊔ L2 L1 ∗ L2 WL (Z/2)n+1 (Z/2)∗(n+1) WL1 ∗ WL2

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Davis complex for a Coxeter group

Right-Angled Coxeter groups

Let L be a finite, flag simplicial complex. W = WL = si ∈ V (L) | s2

i = e, sisj = sjsi iff {si, sj} ∈ E(L)

Examples

L ∆n (∆n)(0) L1 ⊔ L2 L1 ∗ L2 WL (Z/2)n+1 (Z/2)∗(n+1) WL1 ∗ WL2 WL1 × WL2

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Davis complex for a Coxeter group

Right-Angled Coxeter groups

Let L be a finite, flag simplicial complex. W = WL = si ∈ V (L) | s2

i = e, sisj = sjsi iff {si, sj} ∈ E(L)

Examples

L ∆n (∆n)(0) L1 ⊔ L2 L1 ∗ L2 WL (Z/2)n+1 (Z/2)∗(n+1) WL1 ∗ WL2 WL1 × WL2

EW = ΣW = Σ - Davis complex

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Davis complex for a Coxeter group

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Davis complex for a Coxeter group

Example

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Davis complex for a Coxeter group

Example

D∞ = WL where

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Davis complex for a Coxeter group

Example

D∞ = WL where s t L =

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Davis complex for a Coxeter group

Example

D∞ = WL where s t L′ =

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Davis complex for a Coxeter group

Example

D∞ = WL where s t L′ = s t e CL′ =

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Davis complex for a Coxeter group

Example

D∞ = WL where s t L′ = s t e CL′ = ΣD∞ ∼ = R

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Davis complex for a Coxeter group

Example

D∞ = WL where s t L′ = s t e CL′ = ΣD∞ ∼ = R s t e e s t e st s t e s s t e t s t e ts s t e tst

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Davis complex for a Coxeter group

Example

D∞ = WL where s t L′ = s t e CL′ = ΣD∞ ∼ = R s t e e s t e st s t e s s t e t s t e ts s t e tst

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Davis complex for a Coxeter group

Example

D∞ = WL where s t L′ = s t e CL′ = ΣD∞ ∼ = R s t e e s t e st s t e s s t e t s t e ts s t e tst

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Davis complex for a Coxeter group

Example

D∞ = WL where s t L′ = s t e CL′ = ΣD∞ ∼ = R s t e e s t e st s t e s s t e t s t e ts s t e tst

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Davis complex for a Coxeter group

Example

D∞ = WL where s t L′ = s t e CL′ = ΣD∞ ∼ = R s s t e e s t e st s t e s s t e t s t e ts s t e tst

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Davis complex for a Coxeter group

Example

D∞ = WL where s t L′ = s t e CL′ = ΣD∞ ∼ = R s t s t e e s t e st s t e s s t e t s t e ts s t e tst

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Davis complex for a Coxeter group

Example

D∞ = WL where s t L′ = s t e CL′ = ΣD∞ ∼ = R s t s t e e s t e st s t e s s t e t s t e ts s t e tst

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Davis complex for a Coxeter group

Example

D∞ = WL where s t L′ = s t e CL′ = ΣD∞ ∼ = R s t s t e e s t e st s t e s s t e t s t e ts s t e tst ΣWL = WL × CL′/ ∼

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Davis complex for a Coxeter group

Example

D∞ = WL where s t L′ = s t e CL′ = ΣD∞ ∼ = R s t s t e e s t e st s t e s s t e t s t e ts s t e tst ΣWL = WL × CL′/ ∼

Action of W on ΣW

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Davis complex for a Coxeter group

Example

D∞ = WL where s t L′ = s t e CL′ = ΣD∞ ∼ = R s t s t e e s t e st s t e s s t e t s t e ts s t e tst ΣWL = WL × CL′/ ∼

Action of W on ΣW

◮ W ΣW by w · [w′, x] = [ww′, x]

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Davis complex for a Coxeter group

Example

D∞ = WL where s t L′ = s t e CL′ = ΣD∞ ∼ = R s t s t e e s t e st s t e s s t e t s t e ts s t e tst ΣWL = WL × CL′/ ∼

Action of W on ΣW

◮ W ΣW by w · [w′, x] = [ww′, x] ◮ ΣW /W = [e, CL′] = CL′ - strict fundamental domain

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Davis complex for a Coxeter group

Example

D∞ = WL where s t L′ = s t e CL′ = ΣD∞ ∼ = R s t s t e e s t e st s t e s s t e t s t e ts s t e tst ΣWL = WL × CL′/ ∼

Action of W on ΣW

◮ W ΣW by w · [w′, x] = [ww′, x] ◮ ΣW /W = [e, CL′] = CL′ - strict fundamental domain ◮ Stabilisers =

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Davis complex for a Coxeter group

Example

D∞ = WL where s t L′ = s t e CL′ = ΣD∞ ∼ = R s t s t e e s t e st s t e s s t e t s t e ts s t e tst ΣWL = WL × CL′/ ∼

Action of W on ΣW

◮ W ΣW by w · [w′, x] = [ww′, x] ◮ ΣW /W = [e, CL′] = CL′ - strict fundamental domain ◮ Stabilisers = conjugates of subgroups s1, . . . , sn ⊂ W where

{s1, . . . , sn} spans a simplex of L

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Davis complex for a Coxeter group

Example

D∞ = WL where s t L′ = s t e CL′ = ΣD∞ ∼ = R s t s t e e s t e st s t e s s t e t s t e ts s t e tst ΣWL = WL × CL′/ ∼

Action of W on ΣW

◮ W ΣW by w · [w′, x] = [ww′, x] ◮ ΣW /W = [e, CL′] = CL′ - strict fundamental domain ◮ Stabilisers = conjugates of subgroups s1, . . . , sn ⊂ W where

{s1, . . . , sn} spans a simplex of L ⇒ proper action

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Davis complex for a Coxeter group

Example

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Davis complex for a Coxeter group

Example

L s1 s2 s3

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Davis complex for a Coxeter group

Example

L s1 s2 s3 WL ∼ = (Z/2 × Z/2) ∗ Z/2

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Davis complex for a Coxeter group

Example

L s1 s2 s3 WL ∼ = (Z/2 × Z/2) ∗ Z/2 CL′ s1 s2 s3

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Davis complex for a Coxeter group

Example

L s1 s2 s3 WL ∼ = (Z/2 × Z/2) ∗ Z/2 CL′ s1 s2 s3 ΣWL CL′

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Davis complex for a Coxeter group

Example

L s1 s2 s3 WL ∼ = (Z/2 × Z/2) ∗ Z/2 CL′ s1 s2 s3 ΣWL CL′ s1

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Davis complex for a Coxeter group

Example

L s1 s2 s3 WL ∼ = (Z/2 × Z/2) ∗ Z/2 CL′ s1 s2 s3 ΣWL CL′ s1 s2

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Davis complex for a Coxeter group

Example

L s1 s2 s3 WL ∼ = (Z/2 × Z/2) ∗ Z/2 CL′ s1 s2 s3 ΣWL CL′ s1 s2 s3

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Davis complex for a Coxeter group

Example

L s1 s2 s3 WL ∼ = (Z/2 × Z/2) ∗ Z/2 CL′ s1 s2 s3 ΣWL CL′

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Davis complex for a Coxeter group

Theorem (Moussong)

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Davis complex for a Coxeter group

Theorem (Moussong)

ΣW supports a W –invariant CAT(0) metric.

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Davis complex for a Coxeter group

Theorem (Moussong)

ΣW supports a W –invariant CAT(0) metric. Therefore ΣW = EW .

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Davis complex for a Coxeter group

Theorem (Moussong)

ΣW supports a W –invariant CAT(0) metric. Therefore ΣW = EW . dim(ΣWL) = dim(CL′) = dim(L) + 1

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Davis complex for a Coxeter group

Theorem (Moussong)

ΣW supports a W –invariant CAT(0) metric. Therefore ΣW = EW . dim(ΣWL) = dim(CL′) = dim(L) + 1

Example

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Davis complex for a Coxeter group

Theorem (Moussong)

ΣW supports a W –invariant CAT(0) metric. Therefore ΣW = EW . dim(ΣWL) = dim(CL′) = dim(L) + 1

Example

if L = ∆n

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Davis complex for a Coxeter group

Theorem (Moussong)

ΣW supports a W –invariant CAT(0) metric. Therefore ΣW = EW . dim(ΣWL) = dim(CL′) = dim(L) + 1

Example

if L = ∆n then dim(ΣWL) = n + 1

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Davis complex for a Coxeter group

Theorem (Moussong)

ΣW supports a W –invariant CAT(0) metric. Therefore ΣW = EW . dim(ΣWL) = dim(CL′) = dim(L) + 1

Example

if L = ∆n then dim(ΣWL) = n + 1 but WL ∼ = (Z/2)n+1 is finite,

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Davis complex for a Coxeter group

Theorem (Moussong)

ΣW supports a W –invariant CAT(0) metric. Therefore ΣW = EW . dim(ΣWL) = dim(CL′) = dim(L) + 1

Example

if L = ∆n then dim(ΣWL) = n + 1 but WL ∼ = (Z/2)n+1 is finite, so EWL ≃ {pt}.

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Main theorem

Theorem (Petrosyan-P.)

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Main theorem

Theorem (Petrosyan-P.)

There exists a WL–complex BWL (‘Bestvina complex’) such that:

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Main theorem

Theorem (Petrosyan-P.)

There exists a WL–complex BWL (‘Bestvina complex’) such that: 1. BWL and ΣWL are WL–homotopy equivalent

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Main theorem

Theorem (Petrosyan-P.)

There exists a WL–complex BWL (‘Bestvina complex’) such that: 1. BWL and ΣWL are WL–homotopy equivalent Therefore BWL ≃ EWL

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Main theorem

Theorem (Petrosyan-P.)

There exists a WL–complex BWL (‘Bestvina complex’) such that: 1. BWL and ΣWL are WL–homotopy equivalent Therefore BWL ≃ EWL

  • 2. dim(

BWL) = vcdWL

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Main theorem

Theorem (Petrosyan-P.)

There exists a WL–complex BWL (‘Bestvina complex’) such that: 1. BWL and ΣWL are WL–homotopy equivalent Therefore BWL ≃ EWL

  • 2. dim(

BWL) = vcdWL = cdWL

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Main theorem

Theorem (Petrosyan-P.)

There exists a WL–complex BWL (‘Bestvina complex’) such that: 1. BWL and ΣWL are WL–homotopy equivalent Therefore BWL ≃ EWL

  • 2. dim(

BWL) = vcdWL = cdWL (except it could be that cdWL = 2 but dim( BWL) = 3)

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Main theorem

Theorem (Petrosyan-P.)

There exists a WL–complex BWL (‘Bestvina complex’) such that: 1. BWL and ΣWL are WL–homotopy equivalent Therefore BWL ≃ EWL

  • 2. dim(

BWL) = vcdWL = cdWL (except it could be that cdWL = 2 but dim( BWL) = 3) + BWL ‘often’ has a simple cell structure.

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Main theorem

Theorem (Petrosyan-P.)

There exists a WL–complex BWL (‘Bestvina complex’) such that: 1. BWL and ΣWL are WL–homotopy equivalent Therefore BWL ≃ EWL

  • 2. dim(

BWL) = vcdWL = cdWL (except it could be that cdWL = 2 but dim( BWL) = 3) + BWL ‘often’ has a simple cell structure.

Idea

ΣW = W × CL′/ ∼

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Main theorem

Theorem (Petrosyan-P.)

There exists a WL–complex BWL (‘Bestvina complex’) such that: 1. BWL and ΣWL are WL–homotopy equivalent Therefore BWL ≃ EWL

  • 2. dim(

BWL) = vcdWL = cdWL (except it could be that cdWL = 2 but dim( BWL) = 3) + BWL ‘often’ has a simple cell structure.

Idea

ΣW = W × CL′/ ∼ Replace CL′ with a simpler fundamental domain BW and define

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Main theorem

Theorem (Petrosyan-P.)

There exists a WL–complex BWL (‘Bestvina complex’) such that: 1. BWL and ΣWL are WL–homotopy equivalent Therefore BWL ≃ EWL

  • 2. dim(

BWL) = vcdWL = cdWL (except it could be that cdWL = 2 but dim( BWL) = 3) + BWL ‘often’ has a simple cell structure.

Idea

ΣW = W × CL′/ ∼ Replace CL′ with a simpler fundamental domain BW and define

  • BW = W × BW / ∼
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Main theorem

Example

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Main theorem

Example

L s1 s2 s3

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Main theorem

Example

L s1 s2 s3 WL ∼ = (Z/2 × Z/2) ∗ Z/2

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Main theorem

Example

L s1 s2 s3 WL ∼ = (Z/2 × Z/2) ∗ Z/2 CL′ s1 s2 s3

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Main theorem

Example

L s1 s2 s3 WL ∼ = (Z/2 × Z/2) ∗ Z/2 CL′ s1 s2 s3 BW s1, s2 s3

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Main theorem

Example

L s1 s2 s3 WL ∼ = (Z/2 × Z/2) ∗ Z/2 CL′ s1 s2 s3 BW s1, s2 s3 ΣW CL′

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Main theorem

Example

L s1 s2 s3 WL ∼ = (Z/2 × Z/2) ∗ Z/2 CL′ s1 s2 s3 BW s1, s2 s3 ΣW CL′

  • BW

BW

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Main theorem

Example

L s1 s2 s3 WL ∼ = (Z/2 × Z/2) ∗ Z/2 CL′ s1 s2 s3 BW s1, s2 s3 ΣW CL′

  • BW

BW ≃W

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Applications

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Applications

Actions with a strict fundamental domain

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Applications

Actions with a strict fundamental domain

Let a group G act on a simplicial complex X with a strict fundamental domain Y .

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Applications

Actions with a strict fundamental domain

Let a group G act on a simplicial complex X with a strict fundamental domain Y . Then X ∼ = G × Y / ∼

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Applications

Actions with a strict fundamental domain

Let a group G act on a simplicial complex X with a strict fundamental domain Y . Then X ∼ = G × Y / ∼ where ∼ is given in terms of stabilisers of simplices of Y .

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Applications

Actions with a strict fundamental domain

Let a group G act on a simplicial complex X with a strict fundamental domain Y . Then X ∼ = G × Y / ∼ where ∼ is given in terms of stabilisers of simplices of Y . If X = G × Y / ∼ is a model for EG then our theorem applies:

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Applications

Actions with a strict fundamental domain

Let a group G act on a simplicial complex X with a strict fundamental domain Y . Then X ∼ = G × Y / ∼ where ∼ is given in terms of stabilisers of simplices of Y . If X = G × Y / ∼ is a model for EG then our theorem applies: We get a G–complex BG = G × BG/ ∼ with:

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Applications

Actions with a strict fundamental domain

Let a group G act on a simplicial complex X with a strict fundamental domain Y . Then X ∼ = G × Y / ∼ where ∼ is given in terms of stabilisers of simplices of Y . If X = G × Y / ∼ is a model for EG then our theorem applies: We get a G–complex BG = G × BG/ ∼ with:

◮ X ≃G

BG ≃ EG

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Applications

Actions with a strict fundamental domain

Let a group G act on a simplicial complex X with a strict fundamental domain Y . Then X ∼ = G × Y / ∼ where ∼ is given in terms of stabilisers of simplices of Y . If X = G × Y / ∼ is a model for EG then our theorem applies: We get a G–complex BG = G × BG/ ∼ with:

◮ X ≃G

BG ≃ EG

◮ dim(

BG) = cdG (except when cdG = 2)

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Applications

Actions with a strict fundamental domain

Let a group G act on a simplicial complex X with a strict fundamental domain Y . Then X ∼ = G × Y / ∼ where ∼ is given in terms of stabilisers of simplices of Y . If X = G × Y / ∼ is a model for EG then our theorem applies: We get a G–complex BG = G × BG/ ∼ with:

◮ X ≃G

BG ≃ EG

◮ dim(

BG) = cdG (except when cdG = 2)

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Applications

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Applications

Examples

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Applications

Examples

A group G acting properly on a CAT(0) space (simplicial complex).

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Applications

Examples

A group G acting properly on a CAT(0) space (simplicial complex).

◮ Coxeter groups

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Applications

Examples

A group G acting properly on a CAT(0) space (simplicial complex).

◮ Coxeter groups

W ΣW – Davis complex

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Applications

Examples

A group G acting properly on a CAT(0) space (simplicial complex).

◮ Coxeter groups

W ΣW – Davis complex

◮ graph products of finite groups

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Applications

Examples

A group G acting properly on a CAT(0) space (simplicial complex).

◮ Coxeter groups

W ΣW – Davis complex

◮ graph products of finite groups

RACG with si ∼ = Z/2 replaced by arbitrary finite groups Fi

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Applications

Examples

A group G acting properly on a CAT(0) space (simplicial complex).

◮ Coxeter groups

W ΣW – Davis complex

◮ graph products of finite groups

RACG with si ∼ = Z/2 replaced by arbitrary finite groups Fi G acts properly on a right-angled building

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Applications

Examples

A group G acting properly on a CAT(0) space (simplicial complex).

◮ Coxeter groups

W ΣW – Davis complex

◮ graph products of finite groups

RACG with si ∼ = Z/2 replaced by arbitrary finite groups Fi G acts properly on a right-angled building

◮ some automorphism groups of buildings

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Applications

Examples

A group G acting properly on a CAT(0) space (simplicial complex).

◮ Coxeter groups

W ΣW – Davis complex

◮ graph products of finite groups

RACG with si ∼ = Z/2 replaced by arbitrary finite groups Fi G acts properly on a right-angled building

◮ some automorphism groups of buildings

proper, chamber-transitive

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  • THANK YOU
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