The problem of equivariant rigidity is the Γ-homeomorphism classification of Γ-actions on manifolds with compact quotient and with contractible fixed sets for all finite subgroups of Γ. In other words, this is the classification of cocompact EfinΓ-manifolds. We use surgery theory, algebraic K-theory, and the Farrell–Jones Conjecture to give this classification for a family of groups which satis...

We prove that the simplicial boundary of a CAT(0) cube complex admitting a proper, cocompact action by a virtually Z group is isomorphic to the hyperoctahedral triangulation of Sn−1, providing a class of groups G for which the simplicial boundary of a G-cocompact cube complex depends only on G. We also use this result to show that the cocompactly cubulated crystallographic groups in dimension n...

We prove the following: there are infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on hyperbolic space Hn for every n ≤ 19 (resp. n ≤ 6). When n = 7 or 8, they may be taken to be nonarithmetic. Furthermore, for 2 ≤ n ≤ 19, with the possible exceptions n = 16 and 17, the number of essentially distinct Coxeter groups in Hn with noncompact fundamental domain of volume ≤ V gr...

The Jørgensen number of a rank-two non-elementary Kleinian group Γ is J(Γ) = inf{|trX − 4|+ |tr[X,Y ]− 2| : 〈X,Y 〉 = Γ}. Jørgensen’s Inequality guarantees J(Γ) ≥ 1, and Γ is a Jørgensen group if J(Γ) = 1. This paper shows that the only torsion-free Jørgensen group is the figure-eight knot group, identifies all non-cocompact arithmetic Jørgensen groups, and establishes a characterization of coco...

Let M be a convex cocompact, acylindrical hyperbolic 3-manifold of infinite volume, and let M? denote the interior core M. In this paper we show that any geodesic plane in is either closed or dense. We also only countably many planes are closed. These first rigidity theorems for cocompact 3-manifolds volume depend on topology

We give a necessary and sufficient condition for a locally compact group to be isomorphic to a closed cocompact subgroup in the isometry group of a Diestel–Leader graph. As a consequence of this condition, we see that every cocompact lattice in the isometry group of a Diestel–Leader graph admits a transitive, proper action on some other Diestel–Leader graph. We also give some examples of lattic...

We prove geometric superrigidity for actions of cocompact lattices in semisimple Lie groups of higher rank on infinite dimensional Riemannian manifolds of nonpositive curvature and finite telescopic dimension.