## Notes of Samuel Taylor’s Cambridge lecture 9-01-2017

Counting loxodromics

Joint with I. Gekhtman and G. Tiozzo.

1. Introduction

Example. Let ${G}$ be a surface group, acting isometrically on a tree ${X}$. Examples arise from Bass-Serre trees of splittings of the surface along a simple closed curve ${\alpha}$. An element ${g\in G}$ can be elliptic (fixes a vertex) or loxodromic (leaves invariant a geodesic, and translates it by ${\tau(g)=i([g],\alpha)}$.

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 ${(P)}$ is typical if

$\displaystyle \begin{array}{rcl} \frac{|B(n)\cap(P)|}{|B(n)|}\rightarrow 1 \quad\textrm{ as }n\rightarrow\infty. \end{array}$

Question. Let ${G}$ act on some metric space ${X}$. Are loxodromic elements typical ?

Example (where the answer is no). Let ${G=F_2\times F_3}$ with standard generating set. Let ${X=Cay(F_2)}$ on which the ${F_3}$ factor acts trivially.

With respect to a simple random walk, loxodromic is typical. Indeed, the random walk on ${F_3}$ is transient, so trajectories typically have non trivial first component.

However, ${\frac{|B(n)\cap(LOX)|}{|B(n)|}\rightarrow \frac{2}{3}}$.

Therefore we shall stick to hyperbolic groups in the sequel.

2. Hyperbolic setting

Assume ${X}$ is hyperbolic. Loxodromics are isometries ${g}$ such that

$\displaystyle \begin{array}{rcl} \tau(g):=\lim_{n\rightarrow\infty}\frac{1}{n}d(x,g^n x)>0. \end{array}$

We assume that ${G}$ too is hyperbolic, and that the action is non-elementary.

Theorem 2 Let ${G}$ be a hyperbolic group with a nonelementary action on a hyperbolic metric space ${X}$. Then loxodromics are generic.

3. Proof

We study the behaviour of typical geodesics. Geodesics in ${G}$ 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 ${G}$ is a weak limit of uniform measures on balls.

Theorem 3 Let ${G}$ be a hyperbolic group with a nonelementary action on a hyperbolic metric space ${X}$. For a.e. ${\eta\in\partial G}$ (in Patterson-Sullivan measure), and for every geidesic ${(g_n)_n}$ converging to ${\eta}$,

1. ${g_n x}$ converges to a point in ${\partial X}$.
2. $\displaystyle \begin{array}{rcl} \lim_{n\rightarrow\infty}\frac{1}{|g_n|}d(x,g_n x)=L>0, \quad \textrm{ where }L\textrm{ is independent of$

3. There is a quasi-geodesic ray ${r}$ in ${X}$ such that

$\displaystyle \begin{array}{rcl} \lim_{n\rightarrow\infty}\frac{1}{n}d(g_n x,r)=0. \end{array}$

Since displacement satisfies

$\displaystyle \begin{array}{rcl} \tau(g)\geq d(x,gx)-2\langle g(x),g^{-1}(x)\rangle_x+O(\delta), \end{array}$

this implies a stronger statement than in the main thm,

$\displaystyle \begin{array}{rcl} \frac{|\{g\in B(n)\,;\,\tau(g)\geq L|g|\}|}{|B(n)|}\rightarrow 1. \end{array}$

3.1. Technique

We translate typical geodesics into sample paths of a random walk, with Patterson-Sullivan as a hitting measure on ${\partial G}$. And apply the probabilistic result, due to Maher-Tiozzo.

Proposition 4 Given a generating set, there exists finitely many (nonsymmetric, infinite support) probability measures ${\mu_j}$ on ${G}$ such that the Patterson-Sullivan measure

$\displaystyle \begin{array}{rcl} PS=\sum_{g\in G}a_g g_*\nu_{j(g)} \end{array}$

for suitble nonnegative numbers ${a_g}$. ${\nu_j}$ is the hitting measure on ${\partial G}$ for the random walk generated by ${\mu_j}$.

4. Applications

4.1. Translation length in Cayley graph

Apply thm to the action of ${G}$ 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, ${i([g],\alpha)}$ 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 ${\gamma}$ fills iff ${P(\gamma)}$ is pseudo-Anosov.