Compactifications of horospheric products

TL;DR

This paper introduces a new height compactification for horospheric products of infinite trees, providing a complete description and linking it to the Busemann compactification under certain degree conditions.

Contribution

It defines the height compactification for horospheric products and establishes its isomorphism with the Busemann compactification when trees have vertices of degree at least three.

Findings

The height compactification is isomorphic to the Busemann compactification under specified conditions.

Provides a detailed description of Busemann functions in this context.

Discusses applications to ergodic cocycles and their asymptotic behavior.

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 degrees of 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.