Compactifications of horospheric products

Abstract

We define and study a new compactification, called the height compactification of the horospheric product of two infinite trees. We will provide a complete description of this compactification. In particular, we show that this compactification is isomorphic to the Busemann compactification when all the vertices of both trees have degree at least three, which also leads to a precise description of the Busemann functions in terms of the points in the geometric compactification of each tree. We will discuss an application to the asymptotic behavior of integrable ergodic cocycles with values in the isometry group of such horospheric product.The goal of this work is twofold. On the one hand, we will provide a general construction of new compactifications for a metric space X starting from an initial compactification X and a continuous mapping h : X → K, where K is itself a compact metric space. Roughly speaking, this space, called the mapping compactification, is the closure of the diagonal embedding of X in X × K, see Section 2.1 for details. We will then apply this construction to the specific case of horospheric products of trees. This leads to a compactification that, under appropriate conditions, turns out to be isomorphic to the Busemann compactification. Interestingly, these compactifications are both larger than