A GH-compactification of CAT$(0)$-groups via totally disconnected, unimodular actions

TL;DR

This paper characterizes the limits of sequences of proper, geodesically complete CAT(0)-spaces with totally disconnected, unimodular isometry groups, establishing a compactification of such spaces under Gromov-Hausdorff convergence.

Contribution

It provides a detailed description of limits and proves the compactness of the class of metric quotients formed by these spaces and groups, creating a new geometric compactification.

Findings

The class of metric quotients is compact under Gromov-Hausdorff convergence.

The compactification includes locally geodesically complete, locally compact CAT(0)-spaces.

Universal covers are uniformly packed with bounded diameter.

Abstract

We give a detailed description of the possible limits in the equivariant-Gromov-Hausdorff sense of sequences $(X_j,G_j)$, where the $X_j$'s are proper, geodesically complete, uniformly packed, CAT$(0)$-spaces and the $G_j$'s are closed, totally disconnected, unimodular, uniformly cocompact groups of isometries. We show that the class of metric quotients $G/X$, where $X$ and $G$ are as above, is compact under Gromov-Hausdorff convergence. In particular it is a geometric compactification of the class of locally geodesically complete, locally compact, locally CAT$(0)$-spaces with uniformly packed universal cover and uniformly bounded diameter.