## Notes of Camille Horbez’ lecture

Horoboundary of Outer space, and growth under random automorphisms

1. Random growth

Question. Pick an element ${g}$ of free group ${F}$. Apply a sequence of random elements of ${Aut(F)}$. How fast does the length grow after cyclic reduction ?

Theorem 1 Let ${g\in F}$. Let ${\mu}$ be a probability measure on ${Out(F)}$ whose support is finite and generates ${Out(F)}$. Let ${(\Phi_n)}$ be the corresponding random walk on ${Out(F)}$. Then the limit

$\displaystyle \begin{array}{rcl} \lim |\Phi_n(g)|^{1/n}=\lambda>1 \end{array}$

exists almost always.

This is an analogue of Furstenberg’s theorem for ${Sl(N,{\mathbb Z})}$, and of Anders Karlsson for mapping class groups.

2. Oseledec type result

Here is a classical refinement of the above theorems.

Theorem 2 (Furstenberg-Kiefer, Hennion) There is a deterministic filtration ${L_i}$ of ${{\mathbb R}^N}$ and Lyapunov exponents ${\lambda_i}$

My version:

Theorem 3 There is a deterministic tree of subgroups ${H}$ in ${F=F_N}$ and Lyapunov exponents ${\lambda_H}$ such that the growth has rate ${\lambda_H}$ for elements of ${F}$ conjugated into node ${H}$ but in none of its children.

There are at most ${\frac{3N-2}{4}}$ different positive Lyapunov exponents.

3. Horoboundary

This classical tool (Gromov ?) is used in the proof. Let ${X}$ be a (possibly non symmetric) metric space. Map a point ${x\in X}$ to the distance function, up to an additive constant. This maps ${X}$ to ${C(X)/{\mathbb R}}$, equipped with the topology of uniform convergence on compact sets.

Proposition 4 (Walsh) Assume that ${X}$ is geodesic, proper. Then The embedding is a homeomorphism onto its image, whose closure is compact.

Example. Horoboundary of the real line has 2 points.

We apply the following general fact to Outer space.

Theorem 5 (Karlsson-Ledrappier) Asymptotically, the growth of the distance to the origin of a random walk is modelled on the growth of a (random) horofunction ${h}$. I.e., if ${(\Phi_n)}$ is a random walk on a discrete group acting isometrically on ${X}$,

$\displaystyle \begin{array}{rcl} \lim\frac{1}{n}d(x_0,\Phi_n^{-1}(x_0)=\lim\frac{-1}{n}h(\Phi_n^{-1}(x_0)). \end{array}$

4. Outer space

On Outer space (the space of free actions of ${F}$ on trees), we use the Francaviglia-Martino distance (Lipschitz distance). According to White, it is equal to the log of the supremal ratio of translation lengths. It is achieved by an element which is represented, on the quotient graph, by a simple loop, a figure 8 or a pair of glasses. In particular, it is a primitive element. This makes this distance handily computable.

The Ciller-Morgan compactification is obtained when mapping trees to their translation length, viewed as a function on ${F}$, up to rescaling. I modify this construction by restricting to primitive elements of ${F}$, getting what I call the primitive compactification. Elements in the closure are interpreted as isometric actions on ${{\mathbb R}}$-trees.

Theorem 6 The horocompactification of Outer space is homeomorphic to the primitive compactification. This in turn is a proper quotient of the Culler-Morgan compactification.

Example. If orbits of ${F}$ on the real tree ${T}$ are dense, then the equivalence class of ${T}$ is reduced to ${T}$. But this us not always the case. Some equivalence classes are indeed non trivial