Finiteness of CAT(0) group actions

We prove some finiteness results for discrete isometry groups $\mathrm{\Gamma }$ of uniformly packed CAT$(0)$-spaces $X$ with uniformly bounded codiameter (up to group isomorphism), and for CAT$(0)$-orbispaces $M=\mathrm{\Gamma }\mathrm{\backslash }X$ (up to equivariant homotopy equivalence or equivariant diffeomorphism); these results generalize, in nonpositive curvature, classical finiteness theorems of Riemannian geometry. As a corollary, the order of every torsion subgroup of $\mathrm{\Gamma }$ is bounded above by a universal constant only depending on the packing constants and the codiameter. The main tool is a splitting theorem for sufficiently collapsed actions: namely we show that if a geodesically complete, packed, CAT$(0)$-space admits a discrete, cocompact group of isometries with sufficiently small systole then it necessarily splits a non-trivial Euclidean factor.