The horoboundary of outer space, and growth under random automorphisms

TL;DR

This paper characterizes the horoboundary of outer space with the Lipschitz metric, relates it to classical boundaries, and applies this to analyze growth rates of elements under random automorphisms.

Contribution

It provides a detailed description of the horoboundary of outer space and connects it to the classical boundary, enabling new growth rate estimates for random automorphisms.

Findings

The horoboundary is a quotient of the classical boundary with trees identified by homothety.

The set of Busemann points corresponds to trees with dense orbits.

The horoboundary for the backward Lipschitz metric is infinite-dimensional for N≥3.

Abstract

We show that the horoboundary of outer space for the Lipschitz metric is a quotient of Culler and Morgan's classical boundary, two trees being identified whenever their translation length functions are homothetic in restriction to the set of primitive elements of $F_N$. We identify the set of Busemann points with the set of trees with dense orbits. We also investigate a few properties of the horoboundary of outer space for the backward Lipschitz metric, and show in particular that it is infinite-dimensional when $N\ge 3$. We then use our description of the horoboundary of outer space to derive an analogue of a theorem of Furstenberg--Kifer and Hennion for random products of outer automorphisms of $F_N$, that estimates possible growth rates of conjugacy classes of elements of $F_N$ under such products.