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For a given group $G$, it is natural to ask whether one can classify all isometric $G$-actions on Gromov hyperbolic spaces. We propose a formalization of this problem utilizing the complexity theory of Borel equivalence relations. In this…

In this paper, we show the fundamental theorems for rotationally symmetric hypersurfaces, and thus, together with the earlier results in [3] and [4], provide a complete classification of umbilic hypersurfaces in the Heisenberg groups…

There are two main results. The first states that isotropy subgroups of groups acting transitively on a rationally hyperbolic spaces have infinitely generated rational cohomology algebra. Using this fact, we prove that the analogous…

See math.CV/0509030 which replaces this paper.

We conclude the classification of cohomogeneity one actions on symmetric spaces of rank one by classifying cohomogeneity one actions on quaternionic hyperbolic spaces up to orbit equivalence. As a by-product of our proof, we produce…

In this paper we take an approach similar to that in [M] to establish a positive mass theorem for asymptotically hyperbolic spin manifolds admitting corners along a hypersurface. The main analysis uses an integral representation of a…

We prove a topological rigidity result for simple, thick, hyperbolic P-manifolds of dimension 2: isomorphism of the fundamental groups implies homeomorphism of the P-manifolds. An immediate application is a diagram rigidity theorem for…

By considering appropriate finite covering spaces of closed non-orientable surfaces, we construct linear representations of their mapping class group which have finite index image in certain big arithmetic groups.

We define hyperbolic groupoids, generalizing the notion of a Gromov hyperbolic group. Examples of hyperbolic groupoids include actions of Gromov hyperbolic groups on their boundaries, pseudogroups generated by expanding self-coverings,…

Let X be an arbitrary hyperbolic geodesic metric space and let G be a countable non-elementary weakly acylindrical group of isometries of X. We show that the second bounded cohomology group of G with real coefficients or with coefficients…

We study properties of generic elements of groups of isometries of hyperbolic spaces. Under general combinatorial conditions, we prove that loxodromic elements are generic (i.e. they have full density with respect to counting in balls for…

We discuss the linearity and discreteness of amalgamated products of linear word-hyperbolic groups. In particular, we prove that the double of an Anosov group along a maximal cyclic subgroup is always linear, and we construct examples of…

For a closed and orientable surface of genus at least 2, we prove the surface group extensions of the stabilizers of multicurves are hierarchically hyperbolic groups. This answers a question of Durham, Dowdall, Leininger, and Sisto. We also…

In this paper, we investigate some applications of commutator subgroups to homotopy groups and geometric groups. In particular, we show that the intersection subgroups of some canonical subgroups in certain link groups modulo their…

We construct examples of free-by-cyclic hyperbolic groups which fiber in infinitely many ways over Z. The construction involves adding a specialized square 2-cell to a non-positively curved, squared 2-complex defined by labeled oriented…

We study the generalization of the Willmore functional for surfaces in the three-Heisenberg group. Its construction is based on the spectral theory of the Dirac operator coming to the Weierstrass representation of surfaces (see…

We prove that each special Lorentzian holonomy group (with the exception of those including the isotropy groups of K\"ahler symmetric spaces with rank greater than one) can be realized as the holonomy group of a globally hyperbolic…

We show that uniform lattices in some semi-simple groups (notably complex ones) admit Anosov surface subgroups. This result has a quantitative version: we introduce a notion, called $K$-Sullivan maps, which generalizes the notion of…

We establish new results on the weak containment of quasi-regular and Koopman representations of a second countable locally compact group $G$ associated with non-singular $G$-spaces. We deduce that any two boundary representations of a…

We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups holds for the groups of biregular automorphisms of algebraic surfaces. This gives a positive answer to a question of Vladimir L. Popov.