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.


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