Convergence and collapsing of CAT$(0)$-group actions

Nicola Cavallucci, Andrea Sambusetti

Published 2023-04-21Version 1

We study the theory of convergence for groups $\Gamma$ acting geometrically on proper, geodesically complete CAT$(0)$-spaces $X$, and for their quotients $M=\Gamma \backslash X$ (CAT$(0)$-orbispaces). We describe splitting and collapsing phenomena for nonsingular actions, explaining how they occurs and when the limit action is discrete. This leads to finiteness results for nonsingular actions on uniformly packed CAT$(0)$-spaces with uniformly bounded codiameter, which generalize and sharpen, in nonpositive curvature, the classical finiteness theorems of Riemannian geometry. Finally, we prove some closure and compactness theorems in the class of CAT$(0)$-homology orbifolds, and an isolation result for flats.