Finiteness of CAT(0) group actions

We prove some finiteness results for discrete isometry groups Gamma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Gamma$\end{document} of uniformly packed CAT(0)-spaces X with uniformly bounded codiameter (up to group isomorphism), and for CAT(0)-orbispaces M=Gamma\X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M = \Gamma \backslash X$\end{document} (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 Gamma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Gamma$\end{document} 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.