Joint with I. Gekhtman and G. Tiozzo.
Example. Let be a surface group, acting isometrically on a tree . Examples arise from Bass-Serre trees of splittings of the surface along a simple closed curve . An element can be elliptic (fixes a vertex) or loxodromic (leaves invariant a geodesic, and translates it by .
We are interested in the typical behaviour. This can be made precise by the random walk method: this is the behaviour encountered with probability tending to 1 by a random walk.
I prefer the counting method instead.
Definition 1 Fix a Cayley graph. Say a property is typical if
Question. Let act on some metric space . Are loxodromic elements typical ?
Example (where the answer is no). Let with standard generating set. Let on which the factor acts trivially.
With respect to a simple random walk, loxodromic is typical. Indeed, the random walk on is transient, so trajectories typically have non trivial first component.
Therefore we shall stick to hyperbolic groups in the sequel.
2. Hyperbolic setting
Assume is hyperbolic. Loxodromics are isometries such that
We assume that too is hyperbolic, and that the action is non-elementary.
Theorem 2 Let be a hyperbolic group with a nonelementary action on a hyperbolic metric space . Then loxodromics are generic.
We study the behaviour of typical geodesics. Geodesics in can be mapped to paths converging to a point at infinity, bounded paths, unbounded paths wandering around.
The Patterson-Sullivan measure on the boundary of is a weak limit of uniform measures on balls.
Theorem 3 Let be a hyperbolic group with a nonelementary action on a hyperbolic metric space . For a.e. (in Patterson-Sullivan measure), and for every geidesic converging to ,
- converges to a point in .
- There is a quasi-geodesic ray in such that
Since displacement satisfies
this implies a stronger statement than in the main thm,
We translate typical geodesics into sample paths of a random walk, with Patterson-Sullivan as a hitting measure on . And apply the probabilistic result, due to Maher-Tiozzo.
Proposition 4 Given a generating set, there exists finitely many (nonsymmetric, infinite support) probability measures on such that the Patterson-Sullivan measure
for suitble nonnegative numbers . is the hitting measure on for the random walk generated by .
4.1. Translation length in Cayley graph
Apply thm to the action of on its Cayley graph. Get that translation length grows linearly with word length.
4.2. Splittings of surface groups
For a typical element in a surface group, grows linearly with word length.
4.3. Mapping class groups
MCG acts on the curve complex, and loxodromics coincide with pseudo-Anosov classes.
Maher-Rivin: Pseudo-Anosovs are typical in the sense of random walks.
Typicality in counting sense is still open. It holds for irreducible hyperbolic subgroups of MCG
Say a loop fills a surface if the hyperbolic representative divides into disks (i.e. it intersects every essential curve).
Corollary 5 A typical element of a surface group fills the surface.
Indeed, Kra showed that fills iff is pseudo-Anosov.