## Notes of Anastasia Khukhro’s Cambridge lecture, 23-02-2017

Geometry of finite quotients of groups

With Thiebout Delabie.

1. Box spaces

Call a filtration of group ${G}$ a nested sequence of finite index normal subgroups whose intersection is trivial.

Definition 1 The box space of a filtration ${\{N_i\}}$ is the disjoint union of ${G/N_i}$ with metric

1. a Cayley graph metric (for the image of a fixed generating system of ${G}$) on ${G/N_i}$,
2. ${d(G/N_i,G/N_j)=\mathrm{diameter}(G/N_i)+\mathrm{diameter}(G/N_j)}$.

The idea is to study geometric properties of all ${G/N_i}$ uniformly. Changing the generating set of ${G}$ produces a coarsely equivalent box space.

On the other hand, box space depends crucially on the filtration. I construct a filtration such that even subfiltration and odd subfiltration produce drastically different bow spaces (e.g. one is coarsely Hilbert embeddable, the other is an expander).

Some of the geometry of ${G}$ is recovered: every ${R}$-ball in ${G}$ embeds isometrically infinitely many times in the box space. There is more in it.

Proposition 2

1. ${G}$ is amenable iff box space has property A (Guentner).
2. ${G}$ is Kazhdan ${\Rightarrow}$ box space is an expander (Margulis).
3. Polynomial growth of bow space ${\Rightarrow}$ ${G}$ surjects to ${{\mathbb Z}}$ (Breuillard-Tointon).

2. The geometric rigidity question

Rigidity question. If two bow spaces are coarsely equivalent, what can one say of the original groups ?

Analogous to the algebraic rigidity question: does the list of its finite quotients determine a residually finite group ?

Theorem 3 (Khukhro-Valette) Coarse equivalent box spaces ${\Rightarrow}$ quasi-isometric groups.

The converse to our theorem is wrong.

This shows that there are many different expanders, even up to coarse equivalence. Indeed, there are many qi classes of Kazhdan groups (e.g. ${Sl(n,{\mathbb Z})}$ as ${n}$ varies). Using separation profiles, Hume has a quantitative expression of this.

Kajal Das proved that Coarse equivalent box spaces ${\Rightarrow}$ uniformly measure-equivalent groups.

3. Coarse fundamental groups

Question. To what extent can one recover the filtration?

If ${F_S}$ is a free group, ${N}$ a normal subgroup, ${G=F_S/N}$, then ${\pi_1(Cay(G,S))\simeq N}$. So ${N}$ is recoverable, but not from the coarse geometry. We explain a coarse version of the fundamental group, first introduced by Barcelo, Kramer, Laubenbacher, Wearer.

Let ${X}$ a finite Cayley graph. Pick the identity as base point. Let us call paths the 1-Lipschitz maps ${p:\{0,1,\ldots,\ell(p)\}}$ to ${X}$. Say two paths ${p}$ and ${q}$ are ${r}$-close if either

• either ${\ell(p)=\ell(q)}$ and ${\max d(p(i),q(i))\leq r}$,
• or ${\forall i\leq\min\{\ell(p),\ell(q)\}}$, ${p(i)=q(i)}$ and beyond, either ${p(i)=p(\ell(p))}$ or ${q(i)=q(\ell(q))}$.

Say ${p}$, ${q}$ are ${r}$-homotopic if there exists a sequence of paths from ${p}$ to ${q}$, each being ${r}$-close to the next.

Define ${\pi_{1,r}(X,e)}$ as the set of loops based at ${e}$ modulo ${r}$-homotopy. By construction, a ${C}$-quasi-isometry between Cayley graphs of ${G}$ and ${H}$ implies an epimorphism ${\pi_{1,r}(G,e)\rightarrow\pi_{1,Cr+C}(H,e)}$

Theorem 4 (Delabie-Khukhro) Let ${G=\langle S | R\rangle}$ be finitely presented. Let ${K}$ be the max length of relators. Let ${N}$ be a normal subgroup of ${G}$ all of whose nontrivial elements are longer than ${2K}$. Taking ${2K\leq 4r , then ${\pi_{1,r}(Cay(G/N),e)\simeq N}$.

Corollary 5 Coarsely equivalent bow spaces of finitely presented groups ${\Rightarrow}$ isomorphisms between elements of subfiltrations obtained by discarding finitely many elements.

Relies on Hopficity.

What about infinitely presented groups? We have examples of box spaces of wreath products which are coarse equivalent, but with different normal subgroups.

4. Applications

There exist two filtrations of the free group ${F_2}$ such that ${M_i < N_i}$ with bounded index ${[N_i:M_i]}$, and such that the corresponding bow spaces are not coarsely equivalent. In fact, nearly any sequences with rapidly increasing indices would do the job.

There are infinitely many coarse equivalence classes of box spaces of ${F_3}$ which contain Ramanujan graphs (i.e. expanders with asymptotically optimal spectral gap).