Horoboundary of Outer space, and growth under random automorphisms
1. Random growth
Question. Pick an element of free group . Apply a sequence of random elements of . How fast does the length grow after cyclic reduction ?
Theorem 1 Let . Let be a probability measure on whose support is finite and generates . Let be the corresponding random walk on . Then the limit
exists almost always.
This is an analogue of Furstenberg’s theorem for , 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 of and Lyapunov exponents
Theorem 3 There is a deterministic tree of subgroups in and Lyapunov exponents such that the growth has rate for elements of conjugated into node but in none of its children.
There are at most different positive Lyapunov exponents.
This classical tool (Gromov ?) is used in the proof. Let be a (possibly non symmetric) metric space. Map a point to the distance function, up to an additive constant. This maps to , equipped with the topology of uniform convergence on compact sets.
Proposition 4 (Walsh) Assume that 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 . I.e., if is a random walk on a discrete group acting isometrically on ,
4. Outer space
On Outer space (the space of free actions of 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 , up to rescaling. I modify this construction by restricting to primitive elements of , getting what I call the primitive compactification. Elements in the closure are interpreted as isometric actions on -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 on the real tree are dense, then the equivalence class of is reduced to . But this us not always the case. Some equivalence classes are indeed non trivial