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.

4. Joint spectral radius

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.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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