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

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.

arXiv commentThis is the second half of an old preprint. The first part is about finiteness properties of CAT(0) lattices. This second one concerns limits, possibly with collapsing, of CAT(0)-lattices