Actions on products of CAT(-1) spaces

TL;DR

This paper investigates the compactification of the diagonal action of infinite quasi-convex groups on products of proper CAT(-1) spaces, extending understanding of their boundary behavior and group actions.

Contribution

It introduces a maximal open subset of the horofunction compactification for products of CAT(-1) spaces that captures the diagonal group action.

Findings

Identifies a maximal open subset of the horofunction compactification.

Shows this subset compactifies the diagonal action of quasi-convex groups.

Analyzes product actions of hyperbolic groups on different CAT(-1) spaces.

Abstract

We show that for $X$ a proper $\mathrm{CAT}(-1)$ space there is a maximal open subset of the horofunction compactification of $X\times X$ with respect to the maximum metric that compactifies the diagonal action of an infinite quasi-convex group of the isometries of $X$. We also consider the product action of two quasi-convex representations of an infinite hyperbolic group on the product of two different proper $\mathrm{CAT}(-1)$ spaces.