Classifying space for proper actions for groups admitting a strict - PowerPoint PPT Presentation

Classifying space for proper actions Goal Construct possibly simple models for EG of small dimension = vcd G - virtual cohomological dimension cd G - Bredon cohomological dimension Homology H G ∗ ( EG ) ◮ Baum-Connes conjecture ∼ = → K ∗ ( C ∗ K G ∗ ( EG ) − r ( G )) ◮ Atiyah-Segal Completion theorem (L¨ uck-Oliver ’01) Example SL (2 , Z ) � properly by isometries on H 2 ⇒ H 2 ≃ ESL (2 , Z )

Classifying space for proper actions Goal Construct possibly simple models for EG of small dimension = vcd G - virtual cohomological dimension cd G - Bredon cohomological dimension Homology H G ∗ ( EG ) ◮ Baum-Connes conjecture ∼ = → K ∗ ( C ∗ K G ∗ ( EG ) − r ( G )) ◮ Atiyah-Segal Completion theorem (L¨ uck-Oliver ’01) Example SL (2 , Z ) � properly by isometries on H 2 ⇒ H 2 ≃ ESL (2 , Z ) SL (2 , Z ) ∼ = Z / 4 ∗ Z / 2 Z / 6

Classifying space for proper actions Goal Construct possibly simple models for EG of small dimension = vcd G - virtual cohomological dimension cd G - Bredon cohomological dimension Homology H G ∗ ( EG ) ◮ Baum-Connes conjecture ∼ = → K ∗ ( C ∗ K G ∗ ( EG ) − r ( G )) ◮ Atiyah-Segal Completion theorem (L¨ uck-Oliver ’01) Example SL (2 , Z ) � properly by isometries on H 2 ⇒ H 2 ≃ ESL (2 , Z ) SL (2 , Z ) ∼ = Z / 4 ∗ Z / 2 Z / 6 � properly on a tree

Classifying space for proper actions Goal Construct possibly simple models for EG of small dimension = vcd G - virtual cohomological dimension cd G - Bredon cohomological dimension Homology H G ∗ ( EG ) ◮ Baum-Connes conjecture ∼ = → K ∗ ( C ∗ K G ∗ ( EG ) − r ( G )) ◮ Atiyah-Segal Completion theorem (L¨ uck-Oliver ’01) Example SL (2 , Z ) � properly by isometries on H 2 ⇒ H 2 ≃ ESL (2 , Z ) SL (2 , Z ) ∼ = Z / 4 ∗ Z / 2 Z / 6 � properly on a tree ⇒ Tree ≃ ESL (2 , Z )

Davis complex for a Coxeter group

Davis complex for a Coxeter group Right-Angled Coxeter groups

Davis complex for a Coxeter group Right-Angled Coxeter groups Let L be a finite, flag simplicial complex.

Davis complex for a Coxeter group Right-Angled Coxeter groups Let L be a finite, flag simplicial complex.

Davis complex for a Coxeter group Right-Angled Coxeter groups Let L be a finite, flag simplicial complex.

Davis complex for a Coxeter group Right-Angled Coxeter groups Let L be a finite, flag simplicial complex. W = W L = � s i ∈ V ( L ) | s 2 i = e , s i s j = s j s i iff { s i , s j } ∈ E ( L ) �

Davis complex for a Coxeter group Right-Angled Coxeter groups Let L be a finite, flag simplicial complex. W = W L = � s i ∈ V ( L ) | s 2 i = e , s i s j = s j s i iff { s i , s j } ∈ E ( L ) � Examples L W L

Davis complex for a Coxeter group Right-Angled Coxeter groups Let L be a finite, flag simplicial complex. W = W L = � s i ∈ V ( L ) | s 2 i = e , s i s j = s j s i iff { s i , s j } ∈ E ( L ) � Examples ∆ n L W L

Davis complex for a Coxeter group Right-Angled Coxeter groups Let L be a finite, flag simplicial complex. W = W L = � s i ∈ V ( L ) | s 2 i = e , s i s j = s j s i iff { s i , s j } ∈ E ( L ) � Examples ∆ n L ( Z / 2) n +1 W L

Davis complex for a Coxeter group Right-Angled Coxeter groups Let L be a finite, flag simplicial complex. W = W L = � s i ∈ V ( L ) | s 2 i = e , s i s j = s j s i iff { s i , s j } ∈ E ( L ) � Examples ∆ n (∆ n ) (0) L ( Z / 2) n +1 W L

Davis complex for a Coxeter group Right-Angled Coxeter groups Let L be a finite, flag simplicial complex. W = W L = � s i ∈ V ( L ) | s 2 i = e , s i s j = s j s i iff { s i , s j } ∈ E ( L ) � Examples ∆ n (∆ n ) (0) L ( Z / 2) n +1 ( Z / 2) ∗ ( n +1) W L

Davis complex for a Coxeter group Right-Angled Coxeter groups Let L be a finite, flag simplicial complex. W = W L = � s i ∈ V ( L ) | s 2 i = e , s i s j = s j s i iff { s i , s j } ∈ E ( L ) � Examples ∆ n (∆ n ) (0) L 1 ⊔ L 2 L ( Z / 2) n +1 ( Z / 2) ∗ ( n +1) W L

Davis complex for a Coxeter group Right-Angled Coxeter groups Let L be a finite, flag simplicial complex. W = W L = � s i ∈ V ( L ) | s 2 i = e , s i s j = s j s i iff { s i , s j } ∈ E ( L ) � Examples ∆ n (∆ n ) (0) L 1 ⊔ L 2 L ( Z / 2) n +1 ( Z / 2) ∗ ( n +1) W L 1 ∗ W L 2 W L

Davis complex for a Coxeter group Right-Angled Coxeter groups Let L be a finite, flag simplicial complex. W = W L = � s i ∈ V ( L ) | s 2 i = e , s i s j = s j s i iff { s i , s j } ∈ E ( L ) � Examples ∆ n (∆ n ) (0) L 1 ⊔ L 2 L 1 ∗ L 2 L ( Z / 2) n +1 ( Z / 2) ∗ ( n +1) W L 1 ∗ W L 2 W L

Davis complex for a Coxeter group Right-Angled Coxeter groups Let L be a finite, flag simplicial complex. W = W L = � s i ∈ V ( L ) | s 2 i = e , s i s j = s j s i iff { s i , s j } ∈ E ( L ) � Examples ∆ n (∆ n ) (0) L 1 ⊔ L 2 L 1 ∗ L 2 L ( Z / 2) n +1 ( Z / 2) ∗ ( n +1) W L 1 ∗ W L 2 W L 1 × W L 2 W L

Davis complex for a Coxeter group Right-Angled Coxeter groups Let L be a finite, flag simplicial complex. W = W L = � s i ∈ V ( L ) | s 2 i = e , s i s j = s j s i iff { s i , s j } ∈ E ( L ) � Examples ∆ n (∆ n ) (0) L 1 ⊔ L 2 L 1 ∗ L 2 L ( Z / 2) n +1 ( Z / 2) ∗ ( n +1) W L 1 ∗ W L 2 W L 1 × W L 2 W L EW = Σ W = Σ - Davis complex

Davis complex for a Coxeter group

Davis complex for a Coxeter group Example

Davis complex for a Coxeter group Example D ∞ = W L where

Davis complex for a Coxeter group Example s t D ∞ = W L where L =

Davis complex for a Coxeter group Example s t L ′ = D ∞ = W L where

Davis complex for a Coxeter group Example s t s e t L ′ = CL ′ = D ∞ = W L where

Davis complex for a Coxeter group Example s t s e t L ′ = CL ′ = D ∞ = W L where Σ D ∞ ∼ = R

Davis complex for a Coxeter group Example s t s e t L ′ = CL ′ = D ∞ = W L where Σ D ∞ ∼ = R s e t t e s s e t t e s s e t t e s s e st t ts tst

Davis complex for a Coxeter group Example s t s e t L ′ = CL ′ = D ∞ = W L where Σ D ∞ ∼ = R s e t t e s s e t t e s s e t t e s s e st t ts tst

Davis complex for a Coxeter group Example s t s e t L ′ = CL ′ = D ∞ = W L where Σ D ∞ ∼ = R s e t t e s s e t t e s s e t t e s s e st t ts tst

Davis complex for a Coxeter group Example s t s e t L ′ = CL ′ = D ∞ = W L where Σ D ∞ ∼ = R s e t t e s s e t t e s s e t t e s s e st t ts tst

Davis complex for a Coxeter group Example s t s e t L ′ = CL ′ = D ∞ = W L where s Σ D ∞ ∼ = R s e t t e s s e t t e s s e t t e s s e st t ts tst

Davis complex for a Coxeter group Example s t s e t L ′ = CL ′ = D ∞ = W L where s t Σ D ∞ ∼ = R s e t t e s s e t t e s s e t t e s s e st t ts tst

Davis complex for a Coxeter group Example s t s e t L ′ = CL ′ = D ∞ = W L where s t Σ D ∞ ∼ = R s e t t e s s e t t e s s e t t e s s e st t ts tst

Davis complex for a Coxeter group Example s t s e t L ′ = CL ′ = D ∞ = W L where s t Σ D ∞ ∼ = R s e t t e s s e t t e s s e t t e s s e st t ts tst Σ W L = W L × CL ′ / ∼

Davis complex for a Coxeter group Example s t s e t L ′ = CL ′ = D ∞ = W L where s t Σ D ∞ ∼ = R s e t t e s s e t t e s s e t t e s s e st t ts tst Σ W L = W L × CL ′ / ∼ Action of W on Σ W

Davis complex for a Coxeter group Example s t s e t L ′ = CL ′ = D ∞ = W L where s t Σ D ∞ ∼ = R s e t t e s s e t t e s s e t t e s s e st t ts tst Σ W L = W L × CL ′ / ∼ Action of W on Σ W ◮ W � Σ W by w · [ w ′ , x ] = [ ww ′ , x ]

Davis complex for a Coxeter group Example s t s e t L ′ = CL ′ = D ∞ = W L where s t Σ D ∞ ∼ = R s e t t e s s e t t e s s e t t e s s e st t ts tst Σ W L = W L × CL ′ / ∼ Action of W on Σ W ◮ W � Σ W by w · [ w ′ , x ] = [ ww ′ , x ] ◮ Σ W / W = [ e , CL ′ ] = CL ′ - strict fundamental domain

Davis complex for a Coxeter group Example s t s e t L ′ = CL ′ = D ∞ = W L where s t Σ D ∞ ∼ = R s e t t e s s e t t e s s e t t e s s e st t ts tst Σ W L = W L × CL ′ / ∼ Action of W on Σ W ◮ W � Σ W by w · [ w ′ , x ] = [ ww ′ , x ] ◮ Σ W / W = [ e , CL ′ ] = CL ′ - strict fundamental domain ◮ Stabilisers =

Davis complex for a Coxeter group Example s t s e t L ′ = CL ′ = D ∞ = W L where s t Σ D ∞ ∼ = R s e t t e s s e t t e s s e t t e s s e st t ts tst Σ W L = W L × CL ′ / ∼ Action of W on Σ W ◮ W � Σ W by w · [ w ′ , x ] = [ ww ′ , x ] ◮ Σ W / W = [ e , CL ′ ] = CL ′ - strict fundamental domain ◮ Stabilisers = conjugates of subgroups � s 1 , . . . , s n � ⊂ W where { s 1 , . . . , s n } spans a simplex of L

Davis complex for a Coxeter group Example s t s e t L ′ = CL ′ = D ∞ = W L where s t Σ D ∞ ∼ = R s e t t e s s e t t e s s e t t e s s e st t ts tst Σ W L = W L × CL ′ / ∼ Action of W on Σ W ◮ W � Σ W by w · [ w ′ , x ] = [ ww ′ , x ] ◮ Σ W / W = [ e , CL ′ ] = CL ′ - strict fundamental domain ◮ Stabilisers = conjugates of subgroups � s 1 , . . . , s n � ⊂ W where { s 1 , . . . , s n } spans a simplex of L ⇒ proper action

Davis complex for a Coxeter group Example

Davis complex for a Coxeter group Example L s 1 s 3 s 2

Davis complex for a Coxeter group Example W L ∼ L = ( Z / 2 × Z / 2) ∗ Z / 2 s 1 s 3 s 2

Davis complex for a Coxeter group Example W L ∼ L = ( Z / 2 × Z / 2) ∗ Z / 2 s 1 s 3 s 2 CL ′ s 1 s 3 s 2

Davis complex for a Coxeter group Example W L ∼ L = ( Z / 2 × Z / 2) ∗ Z / 2 s 1 s 3 Σ W L s 2 CL ′ s 1 CL ′ s 3 s 2

Davis complex for a Coxeter group Example W L ∼ L = ( Z / 2 × Z / 2) ∗ Z / 2 s 1 s 3 Σ W L s 2 CL ′ s 1 CL ′ s 3 s 2 s 1

Davis complex for a Coxeter group Example W L ∼ L = ( Z / 2 × Z / 2) ∗ Z / 2 s 1 s 3 Σ W L s 2 CL ′ s 1 CL ′ s 2 s 3 s 2 s 1

Davis complex for a Coxeter group Example W L ∼ L = ( Z / 2 × Z / 2) ∗ Z / 2 s 1 s 3 Σ W L s 2 CL ′ s 1 CL ′ s 2 s 3 s 2 s 3 s 1

Davis complex for a Coxeter group Example W L ∼ L = ( Z / 2 × Z / 2) ∗ Z / 2 s 1 s 3 Σ W L s 2 CL ′ s 1 CL ′ s 3 s 2

Davis complex for a Coxeter group Theorem (Moussong)

Davis complex for a Coxeter group Theorem (Moussong) Σ W supports a W –invariant CAT (0) metric.

Davis complex for a Coxeter group Theorem (Moussong) Σ W supports a W –invariant CAT (0) metric. Therefore Σ W = EW .

Davis complex for a Coxeter group Theorem (Moussong) Σ W supports a W –invariant CAT (0) metric. Therefore Σ W = EW . dim (Σ W L ) = dim ( CL ′ ) = dim ( L ) + 1

Davis complex for a Coxeter group Theorem (Moussong) Σ W supports a W –invariant CAT (0) metric. Therefore Σ W = EW . dim (Σ W L ) = dim ( CL ′ ) = dim ( L ) + 1 Example