Special and Extra Special Groups Generalised Bestvina-Brady groups - PowerPoint PPT Presentation

Special and Extra Special Groups Vladimir Vankov Bestvina-Brady groups Special and Extra Special Groups Generalised Bestvina-Brady groups Special Cube Complexes My work Vladimir Vankov 8 February 2019

Special and Extra Outline Special Groups Vladimir Vankov Bestvina-Brady groups Bestvina-Brady groups Generalised Bestvina-Brady groups Special Cube Complexes Generalised Bestvina-Brady groups My work Special Cube Complexes My work

Special and Extra Finiteness Conditions Special Groups Vladimir Vankov Augmentation ideal Bestvina-Brady groups Generalised Bestvina-Brady groups I G ∶= ker Z G → Z Special Cube Complexes My work G finitely generated ⇐ ⇒ I G finitely generated What about finitely presented? G is said to be FP 2 if I G is finitely presented.

Special and Extra Right-Angled Artin Groups Special Groups Vladimir Vankov Bestvina-Brady groups Γ a graph. Denote by Γ 0 the vertices and Γ 1 the edges. Generalised Bestvina-Brady groups Special Cube RAAG ( Γ ) ∶ = ⟨ Γ 0 ∣ [ u , v ] ⇐ Complexes ⇒ uv ∈ Γ 1 ⟩ My work Empty graphs give free groups. Complete graphs give free abelian groups.

Special and Extra Bestvina-Brady Groups Special Groups Vladimir Vankov Bestvina-Brady groups BB L ∶ = ker RAAG ( Γ ) → Z Generalised Bestvina-Brady groups Take L to be the clique complex of Γ (flag simplicial complex). Special Cube Complexes Theorem (M. Bestvina, N. Brady 1997) My work ▸ BB L is finitely presented ⇐ ⇒ π 1 ( L ) is trivial ▸ BB L is FP 2 ⇐ ⇒ H 1 ( L ) is trivial Consider the presentation complex of Higman’s group ⟨ a , b , c , d ∣ a − 1 bab − 2 , b − 1 cbc − 2 , c − 1 dcd − 2 , d − 1 ada − 2 ⟩

Special and Extra Some Geometry Special Groups Vladimir Vankov Bestvina-Brady G L ( S ) : Group of deck transformations of a branched covering of a groups Generalised space made from RAAG ( Γ ) and BB L . Bestvina-Brady groups Special Cube Take the universal cover of the (standard) classifying space of Complexes RAAG ( Γ ) and quotient by BB L . Branching at vertices labelled by My work ’height function’. Branch vertices have pointwise stabilisers in action. Recipe: cover ˜ L → L , subset S ⊂ Z . Presentation is governed by loops in L (more about this on next slide...).

Special and Extra Some algebra Special Groups Vladimir Vankov Bestvina-Brady groups Given a flag simplicial complex L and a subset S ⊂ Z containing 0, Generalised taking ˜ Bestvina-Brady L to be the universal cover of L , let the generators be the groups directed edges of L (with opposites being inverses). The relations in Special Cube Complexes G L ( S ) are now: My work ▸ For each directed triangle ( a , b , c ) in L , the relations abc = 1, a − 1 b − 1 c − 1 = 1 (triangle relations) ▸ Given a finite collection N of loops that normally generate π 1 ( L ) , for each n ∈ S the relation l n 1 l n 2 ... l n k = 1 for each loop ( l 1 ,..., l k ) in N .

Special and Extra Interesting Properties Special Groups Vladimir Vankov Bestvina-Brady groups Generalised ▸ Similar to BB L , has property FP 2 ⇐ Bestvina-Brady ⇒ H 1 ( L ) trivial groups Special Cube ▸ For π 1 ( L ) non-trivial, there are uncountably many G L ( S ) . In Complexes My work fact, Theorem (I. Leary, R. Kropholler, I. Soroko 2018) L finite connected flag complex, not simply connected � ⇒ ∃ uncountably many quasi-isometry classes among G L ( S )

Special and Extra Interesting Properties: Embedding Theorems Special Groups Vladimir Vankov Bestvina-Brady groups Bestvina and Brady showed that not all FP 2 groups are finitely Generalised Bestvina-Brady presented. But their groups all embed into RAAG s. groups Theorem (Ian Leary 2015) Special Cube Complexes Not all FP 2 groups can embed into finitely presented groups. My work Theorem (Higman-Neumann-Neumann 1949) Every countable group embeds in a 2-generated group. Theorem (Ian Leary 2016) Every countable group embeds in a FP 2 group.

Special and Extra Interesting Properties: Relation Modules Special Groups Vladimir Vankov Bestvina-Brady groups For a group G = ⟨ g 1 ,..., g n ∣ r 1 ,..., r m ⟩ ≅ F / R , define Generalised Bestvina-Brady groups Special Cube s ∶ = Rank ( R ab ) = Rank ( R /[ R , R ]) Complexes My work t ∶ = min m ∈ N ∶ ∃ r i ∈ G with ⟨⟨ r 1 ,..., r m ⟩⟩ = R ( t − s ) is called the relation gap. For fixed L , G L ( S ) all have the same R ab . For S = ∅ , the gap is 0. For infinite S , the gap is ∞ . Finite S ?..

Special and Extra Definition: combinatorics of edges in square Special Groups complexes Vladimir Vankov Bestvina-Brady groups Special Cube Complexes ; D . Wise , F . Haglund ( 2007 ) Generalised Bestvina-Brady groups In a square complex, a hyperplane is an parallelism equivalence class Special Cube of edges via being opposite edges in a square. Denote this by ∼ Complexes My work (abuse of notation) u ∼ v ⇐ ⇒ u , v are opposite in a square (same hyperplane) u ↺ v ⇐ ⇒ u , v share a vertex but are not adjacent in any square u ⊥ v ⇐ ⇒ u , v are adjacent in a square (hyperplanes cross)

Special and Extra Definition: hyperplane pathologies Special Groups Vladimir Vankov Bestvina-Brady A square complex is special if it avoids all of the following groups pathologies: (abuse of notation) Generalised Bestvina-Brady groups ▸ u ∼ v and u ⊥ v (hyperplane crosses itself) Special Cube Complexes ▸ orienting each square, u ∼ − u (hyperplane is not 2-sided) My work ▸ u ∼ v and u ↺ v (hyperplane self-osculates) ▸ u ⊥ v and u ↺ v (two hyperplanes inter-osculate) I.e. given any two distinct edges u , v , at most one of the relations ∼ , ↺ , ⊥ holds between [ u ] and [ v ] .

Special and Extra Properties Special Groups Vladimir Vankov Bestvina-Brady groups Generalised Bestvina-Brady groups The main result of interest is Special Cube Complexes Theorem (Wise, Haglund 2008) My work The fundamental group of a special cube complex embeds into SL n ( Z ) . This comes from a map to the classifying space of a RAAG .

Special and Extra Understanding the complex and action Special Groups Using ˜ Vladimir Vankov L → L as the cover of the square by the octagon and taking S = 2 Z , trying to describe the complex on which G L ( S ) acts. Bestvina-Brady groups ▸ Pick basepoint X i per layer using ’height function’ Generalised Bestvina-Brady groups ▸ X i is stabilised by a i b i c i d i . Special Cube Complexes ▸ up- and down-links of X 2 j are squares My work ▸ up- and down-links of X 2 j + 1 are octagons ▸ projecting to 0-layer, X 2 j is the apex of a square-based pyramid whose base spells out the word a 2 j b 2 j c 2 j d 2 j ▸ projecting to 0-layer, X 2 j + 1 is the apex of an octagon-based pyramid whose base spells out the word a 2 j + 1 b 2 j + 1 c 2 j + 1 d 2 j + 1 a 2 j + 1 b 2 j + 1 c 2 j + 1 d 2 j + 1

Special and Extra Description of the complex Special Groups Vladimir Vankov Bestvina-Brady groups a − 1 ⋅ X i + 2 Generalised Bestvina-Brady a − 1 ⋅ v i + 2 a − 1 ⋅ w i + 2 groups Special Cube Complexes My work b i + 1 a i + 1 ba −( i + 1 ) ⋅ X i + 1 X i + 2 X i + 1 u i + 1 v i + 1 a i + 1 X i + 1 a ⋅ X i + 1 w i + 1 a i + 1 ba −( i + 1 ) ⋅ v i + 1 a ⋅ u i + 1 v i + 1 a i + 1 ba − i ⋅ X i a ⋅ X i

Special and Extra Passing to a torsion-free subgroup Special Groups Vladimir Vankov Bestvina-Brady groups Generalised Bestvina-Brady Consider the kernel of a homomorphism from G L ( S ) to a finite groups group. Special Cube Complexes My work First guess: send a to ( 1 , 2 ) . Fails. Hyperplanes need more room not to ’collapse’. Next guess: map to S 8 using 4 disjoint 2-cycles.

Special and Extra Special Special Groups Vladimir Vankov Bestvina-Brady groups Generalised Bestvina-Brady groups Special Cube Complexes My work

Special and Extra Representation Theory Special Groups Vladimir Vankov ’Shortcut’ to proving the complex is special. Bestvina-Brady groups Use faithful representation of the target finite group to study Generalised Bestvina-Brady hyperplane combinatorics. groups Special Cube Complexes Can use techniques from linear algebra to prove that certain My work equations cannot hold � ⇒ avoid hyperplane pathologies. − 1 ⎛ ⎞ ⎛ ⎞ 0 0 0 0 0 1 0 0 0 0 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 1 0 0 0 0 0 1 0 0 0 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ U ∈ ⟨ ⟩ 0 0 1 0 0 0 0 0 1 0 0 0 ⎜ ⎟ ⎜ ⎟ − 1 ⎜ ⎟ ⎜ ⎟ , ⎜ ⎟ ⎜ ⎟ 0 0 0 1 0 0 0 0 0 0 0 ⎜ ⎟ ⎜ ⎟ 0 0 0 0 1 1 0 0 0 0 1 1 ⎝ ⎠ ⎝ ⎠ 0 0 0 0 0 1 0 0 0 0 0 1

Special and Extra Periodic Order Equations in Finite Groups Special Groups Vladimir Vankov Bestvina-Brady groups S -pattern [1,1,1,2,1] Generalised Bestvina-Brady � ∃ a , b , c groups ⇐ Special Cube Complexes o ( a ) = o ( b ) = o ( c ) = 5 My work [ ab , bc ] trivial [ a 2 b 2 , bc ] trivial [ a 3 b 3 , bc ] has order 2.

Special and Extra A terminological coincidence Special Groups Vladimir Vankov Bestvina-Brady groups Generalised Bestvina-Brady groups Special Cube Complexes Extra special groups. My work Central product of finitely many copies of D 8 or Q 8 .