The horoboundary of outer space, and growth under random automorphisms

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.