Involutions of negatively curved groups with wild boundary behavior

We consider pairs (X,Y) where X is a compact, locally CAT(-1) space, and Y is a totally geodesic subspace. The inclusion induces an embedding of the boundaries at infinity of the universal covers; we focus on the case where these are spheres whose dimensions differ by 2. We show that if the embedding is tame, then it is unknotted. We give examples of pairs for which the embedding is knotted (and can be realized as the fixed point set of an involution). We also provide a criterion for knottedness of tame codimension two spheres in high dimensional (>5) spheres. Corresponding results for delta-hyperbolic groups are also discussed. Some corollaries give new results even in the Riemannian manifold case.Comment: 19 page

EZ-structures and topological applications

We introduce the notion of an EZ-structure on a group. Delta-hyperbolic groups and CAT(0)-groups have EZ-structures. We show torsion-free groups having an EZ-structure automatically have an action by homeomorphisms on a closed (high-dimensional) ball, which is well-behaved away from a "bad limit set" in the boundary of the ball. We show that groups having such an action satisfy the Novikov conjecture. For torsion-free delta-hyperbolic groups $G$, we also give a lower bound for the homotopy groups $\pi_n(P(BG))$, where $P$ is the stable topological pseudo-isotopy functor.Comment: 21 pages, final version will appear in Comment. Math. Hel

Finite automorphisms of negatively curved Poincare Duality groups

In this paper, we show that if G is a finite p-group (p prime) acting by automorphisms on a $\delta$-hyperbolic Poincare Duality group, then the fixed subgroup is a Poincare Duality group over Z/p. We also provide examples to show that the fixed subgroup might not even be a Duality group over Z.Comment: 11 pages, 1 figure; revised version has shorter proof for Prop 2.1; appeared in Geom. Funct. Anal. 14 (2004), pgs. 283-29

On the topology of the space of negatively curved metrics

We show that the space of negatively curved metrics of a closed negatively curved Riemannian $n$-manifold, $n\geq 10$, is highly non-connected

Branched covers of hyperbolic manifolds and harmonic maps

We give examples of harmonic maps between negatively curved manifolds with special properties. These negatively curved manifolds do not have the homotopy type of a locally symmetric space

Exotic structures and the limitations of certain analytic methods in geometry

In this survey we review some results concerning negatively curved exotic strucutres (DIFF and PL) and its (unexpected) implications on the limitations of some analytic methods in geometry. This article is dedicated to the memory of Armand Borel

Teichm\"uller Spaces and Bundles with Negatively Curved Fibers

We study the Teichm\"uller space of negatively curved metrics on a high dimensional manifold, with applications to bundles with negatively curved fibers

A caveat on the convergence of the Ricci flow for pinched negatively curved manifolds

We give examples of pinched negatively curved manifolds for which the Ricci flow does not converge smoothly

The Teichm\"{u}ller Space of Pinched Negatively Curved Metrics on a Hyperbolic Manifold is not Contractible

For a smooth manifold $M$ we define the Teichm\"uller space \cT(M) of all Riemannian metrics on $M$ and the Teichm\"uller space \cT^\epsilon(M) of $\epsilon$-pinched negatively curved metrics on $M$, where $0\leq\epsilon\leq\infty$. We prove that if $M$ is hyperbolic the natural inclusion \cT^\epsilon(M)\hookrightarrow\cT(M) is, in general, not homotopically trivial. In particular, \cT^\epsilon(M) is, in general, not contractible

The space of nonpositively curved metrics of a negatively curved manifold

We show that the space of nonpositively curved metrics of a negatively curved manifold is highly non connected