If M is a hyperbolic 3-manifold with a quasigeodesic flow, then we show that 1(M) acts in a natural way on a closed disc by homeomorphisms. Consequently, such a flow either has a closed orbit or the action on the boundary circle is M¨obius-like but not conjugate into PSL(2,R).We conjecture that the latter possibility cannot occur.

We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of multi-linear algebraic structures on the space of smooth functions. In particular, we find algebraic expressions for Weingarten’s formula, the Ricci curvature, and the Codazzi-Mainardi equations. For matrix analogues of embedded surfaces, we define discrete curvatures and Euler characteristics, and a non-commutative GaussBonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of matrices representing the embedding coordinates, and explicit examples are provided. Furthermore, we illustrate the fact that techniques from differential geometry can carry over to matrix analogues by proving that a bound on the discrete Gauss curvature implies a bound on the eigenvalues of the discrete Laplace operator.

We give new examples of hyperbolic and relatively hyperbolic groups of cohomological dimension d for all d ≥ 4 (see Theorem 2.13). These examples result from applying CAT(0)/CAT(−1) filling constructions (based on singular doubly warped products) to finite volume hyperbolic manifolds with toral cusps. The groups obtained have a number of interesting properties, which are established by analyzing their boundaries at infinity by a kind of Morse-theoretic technique, related to but distinct from ordinary and combinatorial Morse theory (see Section 5).

In this note, we discuss the mean curvature flow of graphs of maps between Riemannian manifolds. Special emphasis will be placed on estimates of the flow as a non-linear parabolic system of differential equations. Several global existence theorems and applications to isotopy problems in geometry and topology will be presented. The results are based on joint works of the author with his collaborators I. Medo\v{s}, K. Smoczyk, and M.-P. Tsui.

In a previous paper we have constructed an invariant of fourdimensional manifolds with boundary in the form of an element in the stable homotopy group of the Seiberg-Witten Floer spectrum of the boundary. Here we prove that when one glues two four- manifolds along their boundaries, the Bauer-Furuta invariant of the resulting manifold is obtained by applying a natural pairing to the invariants of the pieces. As an application, we show that the connected sum of three copies of the K3 surface contains no exotic nuclei. In the process we also compute the Floer spectrum for several Seifert fibrations.