Convergence and collapsing of CAT$(0)$-lattices

TL;DR

This paper investigates the convergence behavior of CAT(0)-lattices and their quotients, revealing phenomena of splitting and collapsing, and establishing a compactness theorem with applications to orbifolds and entropy.

Contribution

It introduces a detailed analysis of convergence and degeneration in CAT(0)-lattices, including new compactness results and applications to orbifold geometry.

Findings

Describes splitting and collapsing phenomena in CAT(0)-lattices.

Proves a compactness theorem for CAT(0)-homology orbifolds.

Provides applications such as an entropy-pinching theorem.

Abstract

We study the theory of convergence for CAT$(0)$-lattices (that is groups $\Gamma$ acting geometrically on proper, geodesically complete CAT$(0)$-spaces) and their quotients (CAT$(0)$-orbispaces). We describe some splitting and collapsing phenomena, explaining precisely how these action can degenerate to a possibly non-discrete limit action. Finally, we prove a compactness theorem for the class of compact CAT$(0)$-homology orbifolds, and some applications: an isolation result for flat orbispaces and an entropy-pinching theorem.