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

Special and Extra Special Groups

Vladimir Vankov 8 February 2019

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

Outline

Bestvina-Brady groups Generalised Bestvina-Brady groups Special Cube Complexes My work

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

Finiteness Conditions

Augmentation ideal IG ∶= ker ZG → Z G finitely generated ⇐ ⇒ IG finitely generated What about finitely presented? G is said to be FP2 if IG is finitely presented.

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

Right-Angled Artin Groups

Γ a graph. Denote by Γ0 the vertices and Γ1 the edges. RAAG(Γ) ∶= ⟨Γ0 ∣ [u,v] ⇐ ⇒ uv ∈ Γ1⟩ Empty graphs give free groups. Complete graphs give free abelian groups.

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

Bestvina-Brady Groups

BBL ∶= ker RAAG(Γ) → Z Take L to be the clique complex of Γ (flag simplicial complex).

Theorem (M. Bestvina, N. Brady 1997)

▸ BBL is finitely presented ⇐ ⇒ π1(L) is trivial ▸ BBL is FP2 ⇐ ⇒ H1(L) is trivial Consider the presentation complex of Higman’s group ⟨a,b,c,d ∣ a−1bab−2,b−1cbc−2,c−1dcd−2,d−1ada−2⟩

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

Some Geometry

GL(S): Group of deck transformations of a branched covering of a space made from RAAG(Γ) and BBL. Take the universal cover of the (standard) classifying space of RAAG(Γ) and quotient by BBL. Branching at vertices labelled by ’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 Special Groups Vladimir Vankov Bestvina-Brady groups Generalised Bestvina-Brady groups Special Cube Complexes My work

Some algebra

Given a flag simplicial complex L and a subset S ⊂ Z containing 0, taking ˜ L to be the universal cover of L, let the generators be the directed edges of L (with opposites being inverses). The relations in GL(S) are now: ▸ For each directed triangle (a,b,c) in L, the relations abc = 1, a−1b−1c−1 = 1 (triangle relations) ▸ Given a finite collection N of loops that normally generate π1(L), for each n ∈ S the relation ln

1 ln 2 ...ln k = 1 for each loop

(l1,...,lk) in N.

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

Interesting Properties

▸ Similar to BBL, has property FP2 ⇐ ⇒ H1(L) trivial ▸ For π1(L) non-trivial, there are uncountably many GL(S). In fact,

Theorem (I. Leary, R. Kropholler, I. Soroko 2018)

L finite connected flag complex, not simply connected

• ⇒ ∃ uncountably many quasi-isometry classes among GL(S)

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

Interesting Properties: Embedding Theorems

Bestvina and Brady showed that not all FP2 groups are finitely

• presented. But their groups all embed into RAAGs.

Theorem (Ian Leary 2015)

Not all FP2 groups can embed into finitely presented groups.

Theorem (Higman-Neumann-Neumann 1949)

Every countable group embeds in a 2-generated group.

Theorem (Ian Leary 2016)

Every countable group embeds in a FP2 group.

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

Interesting Properties: Relation Modules

For a group G = ⟨g1,...,gn ∣ r1,...,rm⟩ ≅ F/R, define s ∶= Rank(Rab) = Rank(R/[R,R]) t ∶= min

m∈N ∶ ∃ri ∈ G with ⟨⟨r1,...,rm⟩⟩ = R

(t − s) is called the relation gap. For fixed L, GL(S) all have the same Rab. For S = ∅, the gap is 0. For infinite S, the gap is ∞. Finite S?..

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

Definition: combinatorics of edges in square complexes

Special Cube Complexes; D. Wise,F. Haglund (2007) In a square complex, a hyperplane is an parallelism equivalence class

• f edges via being opposite edges in a square. Denote this by ∼

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

Definition: hyperplane pathologies

A square complex is special if it avoids all of the following pathologies: (abuse of notation) ▸ u ∼ v and u ⊥ v (hyperplane crosses itself) ▸ orienting each square, u ∼ −u (hyperplane is not 2-sided) ▸ 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 Special Groups Vladimir Vankov Bestvina-Brady groups Generalised Bestvina-Brady groups Special Cube Complexes My work

Properties

The main result of interest is

Theorem (Wise, Haglund 2008)

The fundamental group of a special cube complex embeds into SLn(Z). This comes from a map to the classifying space of a RAAG.

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

Understanding the complex and action

Using ˜ L → L as the cover of the square by the octagon and taking S = 2Z, trying to describe the complex on which GL(S) acts. ▸ Pick basepoint Xi per layer using ’height function’ ▸ Xi is stabilised by aibicidi. ▸ up- and down-links of X2j are squares ▸ up- and down-links of X2j+1 are octagons ▸ projecting to 0-layer, X2j is the apex of a square-based pyramid whose base spells out the word a2jb2jc2jd2j ▸ projecting to 0-layer, X2j+1 is the apex of an octagon-based pyramid whose base spells out the word a2j+1b2j+1c2j+1d2j+1a2j+1b2j+1c2j+1d2j+1

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

Description of the complex

vi+1 a ⋅ ui+1 vi+1 ui+1 ai+1 a ⋅ Xi Xi+1 a ⋅ Xi+1 Xi+2 wi+1 ai+1ba−(i+1) ⋅ vi+1 a−1 ⋅ wi+2 a−1 ⋅ vi+2 bi+1 ai+1ba−i ⋅ Xi Xi+1 ai+1ba−(i+1) ⋅ Xi+1 a−1 ⋅ Xi+2

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

Passing to a torsion-free subgroup

Consider the kernel of a homomorphism from GL(S) to a finite group. First guess: send a to (1,2). Fails. Hyperplanes need more room not to ’collapse’. Next guess: map to S8 using 4 disjoint 2-cycles.

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

Special

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

Representation Theory

’Shortcut’ to proving the complex is special. Use faithful representation of the target finite group to study hyperplane combinatorics. Can use techniques from linear algebra to prove that certain equations cannot hold ⇒ avoid hyperplane pathologies. U ∈ ⟨ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ −1 1 1 1 1 1 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 1 1 −1 1 1 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⟩

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

Periodic Order Equations in Finite Groups

S-pattern [1,1,1,2,1] ⇐ ∃a,b,c

• (a) = o(b) = o(c) = 5

[ab,bc] trivial [a2b2,bc] trivial [a3b3,bc] has order 2.

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

A terminological coincidence

Extra special groups. Central product of finitely many copies of D8 or Q8.

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

Thank you

Thanks for listening!