Qayum Khan (University of Notre Dame)
Rigidity of pseudo-free group actions on contractible manifolds
Abstract: Joint with Frank Connolly (U Notre Dame) and Jim Davis (Indiana U). We discuss Quinn's equivariant generalization of the Borel Conjecture. This concerns cocompact proper actions of a discrete group $\Gamma$ on a Hadamard manifold $X$. We give a complete solution when the action of $\Gamma$ is pseudo-free and when $X$ more generally is a $\mathrm{CAT}(0)$ manifold. Here, pseudo-free means that the singular set is discrete. A rich class of examples is obtained from crystallographic groups $\Gamma$ made out of isometric spherical space form groups $G$. If $\Gamma$ has no elements of order two, then we obtain equivariant topological rigidity of the pair $(X, \Gamma)$. Hence, if $\Gamma$ is torsion-free, then we generalize a recent theorem of A. Bartels and W. Lück, which validates the classical Borel Conjecture for $\mathrm{CAT}(0)$ fundamental groups. Otherwise, if $\Gamma$ has elements of order two, we show how to parameterize all possible counter-examples, in terms of Cappell's $\mathrm{UNil}$ summands of the L-theory of infinite dihedral groups. In certain cases, these are detected along hypersurfaces in the orbifold $X / \Gamma$ by generalized Arf invariants. |
