## Notes of Indira Chatterji’s Southampton lecture 27-03-2017

Acylindrical hyperbolicity and ${CAT(0)}$-cube complexes

Joint with Alexandre Martin.

Theorem 1 Let ${G}$ act on a finite dimensional ${CAT(0)}$ cube complex ${X}$. Assume action is essential and non-elementary. Assume that there exist two hyperplanes of ${X}$ the common stabilizer is finite. Then ${G}$ is acylindrically hyperbolic (AH).

Corollary 2 Artin groups of FC type and defining graphs of diameter ${\geq 3}$ are AH.

This generalizes the corresponding result of Minasyan-Osin for trees.

1. Acylindrical hyperbolicity

Say an isometric action is acylindrical if, for far enough points, the number of group elements moving both points a bounded amount is bounded. I.e., ${\forall r\geq 0}$, ${\exists L(r)}$, ${\exists N(r)}$ such that for all ${x}$, ${y\in X}$ with ${d(x,y)\geq L(r)}$,

$\displaystyle \begin{array}{rcl} |\{g\in G\,;\,d(x,gx)

Definition 3 Say ${G}$ is acylindrically hyperbolic if it is not virtually cyclic and admits an acylindrical action on a hyperbolic metric space, with at least one unbounded orbit.

This arose from Osin (Bowditch)’s classification of acylindrical actions on hyperbolic spaces: either orbits are bounded, or group is virtually cyclic with a loxodromic element, or there are infinitely many loxodromic elements (loxodromic means orbit map is a quasi-isometric embedding).

The notion was developped my many people simultaneously. It has plenty of consequences. There are many nice examples.

1.1. Examples

${Out(F_n)}$ (Bestvina-Feighn), ${MCG}$ (Mazur-Minsky and Bowditch), RAAG’s (Sisto). Relatively hyperbolic groups.

Non-examples: products, ${SL_n({\mathbb Z})}$, ${n\geq 3}$.

1.2. Properties

SQ-universality (Dahmani-Guirardel-Osin). Infinite bounded cohomology (Bestvina-Bramberg-Fujiwara and Hamenstadt). ${C^*}$-simple (Osin). Property Pnaive (given 2 element, can find a third to play ping-pong).

Question. Can a AH group have Property F${\ell^p}$ for all ${p}$?

2. Martin’s criterion

Let ${G}$ act on ${X}$. Say an element ${g\in G}$ is an ubercontraction if it admits a system of checkpoints, i.e. there exists a bounded set ${S\subset X}$ (set ${S_j=g^j S}$) and a function ${f:\bigcup_j S_j\rightarrow{\mathbb R}}$ such that ${f(x')}$, ${f(y')}$ belong to different unbounded components of ${{\mathbb R}\setminus Im(S_i)}$, then every geodesic from ${x}$ to ${y}$ go through ${S_i}$.

Assume that for every ubercontraction ${g}$, there is an ${m0}$, such that for all ${m\geq m_0}$, the common statibilizer of ${S}$ and ${g^m S}$ is finite. Then ${G}$ is either virtually cyclic or AH.

Note that ${X}$ is not assumed to be hyperbolic.

3. ${CAT(0)}$ cube complexes

Uberseparation of hyperplanes means that not only are hyperplanes disjoint, but all hyperplanes cossing one and all hyperplanes crossing the other are disjoint. Caprace-Sageev show that, in presence of an essential non-elementary isometric group action, uberseparated hyperplanes always exist. Uberseparated hyperplane have some kind of shortest bridge that is crossed by every geodesic joining them. Hence Martin’s criterion applies.

4. RAAG’s

The datum ${\Gamma}$ is an edge-labelled graph, where labels are integers ${\geq 2}$. The presentation has a generator per vertex, and a relator ${(sts)\cdots(sts)=(tst)\cdots(tst)}$ per edge. Charney-Davis construct an isometric action of the resulting group ${A_\Gamma}$ on a ${CAT(0)}$ cube complex.

Say that ${A_\Gamma}$ is FC if every finite complete subgraph generates a special subgroup of finite type.

Is the class of groups to which the theorem applies strictly larger than what covers the special case of trees? Yes. Up to commensurabilty? Hopefully yes.