## Notes of Aditi Kar’s Cambridge lecture 09-05-2017

Ping-pong on ${CAT(0)}$ cube complexes

Joint work with Michah Sageev.

Sageev and Wise’s version for ${CAT(0)}$ cubed groups states that a subgroup of such a group is either finitely generated virtually abelian, or contains a free group (this requires an extra assumption: there is a bound on the size of finite subgroups). We study variants of this.

1. Property Pnaive

A group ${G}$ has property Pnaive if for every finite subset ${F}$ not containng 1, there exists a element ${y\in G}$ of infinite order such ${\langle x,\,x\in F,y\rangle =\langle x,\,x\in F,\rangle\star \langle y\rangle}$.

This was introduced by Bekka-Cowling-de la Harpe in connection with ${C^*}$-simplicity. ${G}$ is ${C^*}$-simple if ${C_r^*(G)}$ has no nontrivial ideals. They prove that Pnaive implies ${G}$ is ${C^*}$-simple.

Unknown wether ${Sl(n,{\mathbb Z})}$, ${n\geq 3}$ have Pnaive.

1.1. A book-keeping theorem

Theorem 1 For a group ${G}$ acting properly and cocompactly on a finite dimensional ${CAT(0)}$ cube complex ${X}$, the following are equivalent:

1. ${G}$ has Pnaive.
2. ${G}$ is ${C^*}$-simple.
3. ${G}$ is ICC.
4. The amenable radical of ${G}$ is trivial.
5. the ${G}$-action is faithful and ${X}$ is not Euclidean.

Due to many people. Our contribution is (5)${\Rightarrow}$(1).

If one relaxes the cocompactness assumption, and replaces it with a bound on the size of finite subgroups, (2)${\Leftrightarrow}$(4) remains true (Breuillard-K-K-Ozawa).

1.2. Main theorem

Say a subgroup is inessential if there is an orbit which is entirely contanined in some halfspace. Say a family of subgroups is simultaneously inessential if this happens for the whole family with the same halfspace.

Theorem 2 (Kar-Sageev) Let ${G}$ act properly and cocompactly on a finite dimensional ${CAT(0)}$ cube complex ${X}$, which is irreducible, not Euclidean. Let ${A_i}$ be simultaneously inessential subgroups. Then there exists ${y\in G}$ such that ${\langle A_i,y\rangle=A_i\star\langle y\rangle}$.

2. Uniform exponential growth

It does not follow from earlier statements, by lack of uniformity.

See Sageev’s talk at NPC in action january 2017 workshop. For groups acting freely on square complexes, we show that there are short elements generating a free subgroup.

3. Proofs

We use the Roller boundary, obtained from ultrafilters on the collection of all half-spaces.

Say two hyperplanes are strongly separated if no hyperplane intersecting one of them may intersect the other. Caprace and Sageev show that there exist chains of halfspaces in which consecutive hyperplanes are strongly separated. The corresponding ultrafilters are called strongly separated. We work with the closure ${S}$ of strongly separated ultrafilters in the Roller boundary.

Here are the main properties used: the action of ${G}$ on ${S}$ is topologically free. Three distinct points of ${S}$ have a well-defined median which is a vertex of ${X}$. The fixed-point set of an isometry on ${S}$ has empty interior.

4. Questions

Lubotzky: what about mapping class groups? Johanna Manganas shows uniform exponential growth. Bridson-de la Harpe prove ${C^*}$-simplicity. Abbot-Dahmani prove Pnaive for acylindrically hyperbolic groups.