## Notes of Emmanuel Breuillard’s Cambridge workshop lecture 12-05-2017

How to quickly generate a nice hyperbolic element?

Joint work with Koji Fujiwara, done here in Cambridge during this program. Although we started thinking of it long ago.

Let ${G}$ be a group, ${S}$ a finite set. How large must ${n}$ be in order that knowledge of ${S^n}$ implies knowledge of ${\Gamma:=\langle S \rangle}$?

If ${\Gamma}$ contains nice elements, do they already occur in ${S^n}$?

If ${\Gamma}$ has fast growth, does there exist ${g\in S}$ whose powers grow as fast?

1. Hilbert 5th problem

In Gleason’s 1952 proof that NSS groups are Lie groups, the following quantity arises: for ${U}$ a neighborhood of identity,

$\displaystyle \begin{array}{rcl} n_U(S)=\inf\{n\,;\,S^n\not\subset U\}. \end{array}$

Gleason’s Lemma states that there exists ${g\in S}$ such that

$\displaystyle \begin{array}{rcl} n_U(g)\leq C\,n_U(S). \end{array}$

Yamabe extended this into a trapping lemma.

2. Groups acting on trees

Serre’s lemma: given tree isometries ${a}$ and ${b}$ without common fixed point, at least one of ${a}$, ${b}$ or ${ab}$ has no fixed point at all.

It follows that finitely generated torsion groups have a global fixed point on trees.

An analogous result holds in ${Sl(2,{\mathbb C})}$: if ${a}$, ${b}$ and ${[a,b]}$ are elliptic, then ${ab}$ is hyperbolic.

3. Escaping elliptic elements

In a Euclidean isometry group, if all elements are elliptic, does ${\Gamma}$ have a global fixed points? No! Bass gave a counterexample in ${{\mathbb R}^4={\mathbb C}^2}$. He takes two complex affine motions whose linear parts generate elements of ${SU(2)}$ none of whose has eigenvalue 1.

However, it is the case in irreducible symmetric spaces: there exists a global fixed point in the space or on its boundary. The proof is algebraic, and non-quantitative. In Euclidean buildings, Anne Parreau shows that there must be a fixed point in the building itself.

Does there exists ${n}$ such that if ${\Gamma}$ contains a non-elliptic element, then there is one already in ${S^n}$?

It turns out that answer is no for symmetric spaces but yes for Bruhat-Tits buildings.

3.1. Infinite order

Non-elliptic and infinite order behave differently.

Theorem 1 (Effective Schur’s Lemma) There exists ${n=n(d)}$ such that if ${\gamma\subset Gl(d,{\mathbb C})}$ generates an infinite subgroup, then ${S^n}$ contains an element of infinite order.

The proof uses equidistribution of Galois orbits. ${n(d)}$ must tend to infinity with ${d}$ (Bartholdi-Cornulier, de la Harpe).

3.2. Counterexample

Start with elements in ${K}$ generating a free subgroup such that no element but identity has eigenvalue 1 (this exists by Baire and Borel/Larsen’s theorem on density of images of word maps). Perturb them a bit in ${G}$.

If elliptic elements are not Zariski-dense, a bound exists. It follows from Eskin-Mozes-Oh’s escape from subvarieties theorem.

Rota and Strang 1960: for a finite set ${S}$ of matrices, joint spectral radius is

$\displaystyle \begin{array}{rcl} R(S)=\lim_{n\rightarrow\infty}(\max_{g\in S^n}|g|)^{1/n} \end{array}$

It is important in applied maps (wavelets…).

Berger-Wang: can replace norms with eigenvalues.

Note that ${R(S)=1}$ iff ${\Gamma}$ is made of elliptic elements.

Bochi 2002: the Berger-Wang theorem holds in finite time. There are ${k}$ and ${c}$ depending on dimension only such that

$\displaystyle \begin{array}{rcl} R(S)\geq \sup_{n\leq n(d)}(\max_{g\in S^n}|\lambda(g)|)^{1/n}\geq c\, R(S). \end{array}$

The proof is algebraic.

5. Geometric point of view

I want to give a geometric proof of some of the above results.

On a metric space, a set of isometries has a joint minimal displacement

$\displaystyle \begin{array}{rcl} L(S):=\inf_{x\in X}\max_{x\in S}d(x,sx). \end{array}$

and an asymptotic minimal displacement

$\displaystyle \begin{array}{rcl} \ell(S):=\lim \frac{1}{n}L(S^n). \end{array}$

When ${S}$ has only one element, ${\ell(g)}$ equals translation length.

Set

$\displaystyle \begin{array}{rcl} \lambda(S)=\max_{s\in S}\lambda(s), \quad\textrm{and}\quad \lambda(S)=\frac{1}{k}\max_{s\in S^k}\ell(s). \end{array}$

Berger-Wang generalizes.

Question. When does equality ${\lambda_\infty(S)=\ell(S)}$ hold?

For affine isometric actions on Hilbert spaces, vanishing of ${L}$ and ${\ell}$ correspond to vanishing of reduced cohomology,…

5.1. Results

Equality ${\lambda_\infty(S)=\ell(S)}$ holds for buildings, symmetric spaces and hyperbolic spaces.

Theorem 2 (Breuillard-Fujiwara) For buildings, there is ${k=O(dim(X))}$ such that equality ${\lambda_k(S)=\ell(S)}$ holds.

For symmetric spaces there are ${k,C=O(dim(X))}$ such that inequalities ${\ell(S)-C\leq\lambda_k(S)\leq\ell(S)}$ holds.

For ${\delta}$-hyperbolic metric spaces, there is ${C=C(\delta)}$ such that ${\ell(S)-C\leq\lambda_2(S)\leq\ell(S)}$.

Last statement generalizes Serre’s lemma. Our proof is a quasification of Serre’s. The second generalizes Bochi’s. Our proof is not fully geometric, it still uses some linear algebra.

It follows that if ${L(S)>0}$, there is an element of length ${k}$ which is responsible for that.

5.2. Proof

I stress how useful Helly’s theorem is: in a ${d}$-dimensional ${CAT(0)}$ space, if convex subsets have non-empty ${d+1}$-wise intersections, then the intersection of the whole family is non-empty.

There are even more general versions. There is a ${\delta}$-hyperbolic version.

It implies that ${L(S)=\sup\{L(S')\,;\,S'\subset S,\,|S'|\leq d+1\}}$.

5.3. Application to uniform exponential growth

Corollary: for a ${\delta}$-hyperbolic space, either ${L(S)\leq C\,\delta}$ (i.e. ${S}$ almost fixes a point) or two elements generating a free semi-group can be found in ${S^3}$.

Note that the second case may never occur (e.g. Burnside groups).

This improves on Besson-Courtois-Gallot 2011. They obtained exponential growth for pinched Riemannian manifolds, but could not find a free semi-group, due to possible elliptics.

We recover their result because almost fixed points are ruled out by Margulis Lemma.

In hyperbolic spaces, a version of Margulis lemma follows for the structure theorem for approximate groups. Therefore we can state:

Corollary: for a ${\delta}$-hyperbolic space in which every ball of radius ${2\delta}$ is covered by ${K}$ balls of radius ${\delta}$, a free sub-semigroup is generated by elements of length ${\leq N(K)}$ independent on ${\delta}$.

Hope to cover mapping class groups.