Finiteness of CAT(0) group actions

We prove some finiteness results for discrete isometry groups of uniformly packed CAT(0)-spaces X with uniformly bounded codiameter (up to group isomorphism), and for CAT(0)-orbispaces (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 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.