** Counting loxodromics **

Joint with I. Gekhtman and G. Tiozzo.

**1. Introduction **

**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 1Fix 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.

However, .

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 2Let be a hyperbolic group with a nonelementary action on a hyperbolic metric space . Then loxodromics are generic.

**3. Proof **

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 3Let 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,

** 3.1. Technique **

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 4Given 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. Applications **

** 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 5A typical element of a surface group fills the surface.

Indeed, Kra showed that fills iff is pseudo-Anosov.