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 be a group, a finite set. How large must be in order that knowledge of implies knowledge of ?
If contains nice elements, do they already occur in ?
If has fast growth, does there exist 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 a neighborhood of identity,
Gleason’s Lemma states that there exists such that
Yamabe extended this into a trapping lemma.
2. Groups acting on trees
Serre’s lemma: given tree isometries and without common fixed point, at least one of , or 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 : if , and are elliptic, then is hyperbolic.
3. Escaping elliptic elements
In a Euclidean isometry group, if all elements are elliptic, does have a global fixed points? No! Bass gave a counterexample in . He takes two complex affine motions whose linear parts generate elements of 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 such that if contains a non-elliptic element, then there is one already in ?
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 such that if generates an infinite subgroup, then contains an element of infinite order.
The proof uses equidistribution of Galois orbits. must tend to infinity with (Bartholdi-Cornulier, de la Harpe).
Start with elements in 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 .
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 of matrices, joint spectral radius is
It is important in applied maps (wavelets…).
Berger-Wang: can replace norms with eigenvalues.
Note that iff is made of elliptic elements.
Bochi 2002: the Berger-Wang theorem holds in finite time. There are and depending on dimension only such that
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
and an asymptotic minimal displacement
When has only one element, equals translation length.
Question. When does equality hold?
For affine isometric actions on Hilbert spaces, vanishing of and correspond to vanishing of reduced cohomology,…
Equality holds for buildings, symmetric spaces and hyperbolic spaces.
Theorem 2 (Breuillard-Fujiwara) For buildings, there is such that equality holds.
For symmetric spaces there are such that inequalities holds.
For -hyperbolic metric spaces, there is such that .
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 , there is an element of length which is responsible for that.
I stress how useful Helly’s theorem is: in a -dimensional space, if convex subsets have non-empty -wise intersections, then the intersection of the whole family is non-empty.
There are even more general versions. There is a -hyperbolic version.
It implies that .
5.3. Application to uniform exponential growth
Corollary: for a -hyperbolic space, either (i.e. almost fixes a point) or two elements generating a free semi-group can be found in .
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 -hyperbolic space in which every ball of radius is covered by balls of radius , a free sub-semigroup is generated by elements of length independent on .
Hope to cover mapping class groups.