The Cannon Conjecture from the geometric group theory asserts that a word hyperbolic group that acts effectively on its boundary, and whose boundary is homeomorphic to the 2-sphere, is isomorphic to a Kleinian group. We prove the following Criterion for Cannon’s Conjecture: A hyperbolic group G (that acts effectively on its boundary) whose boundary is homeomorphic to the 2-sphere is isomorphic ...

Based on Spiridonov's analysis of elliptic generalizations of the Gauss hypergeomet-ric function, we develop a common framework for 7-parameter families of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric functions. In each case we derive the symmetries of the generalized hypergeometric function under the Weyl group of type E 7 (ellip-tic, hyperbolic) and of type E 6...

Let G be a group which is hyperbolic relative to a collection of subgroups H1, and it is also hyperbolic relative to a collection of subgroups H2. Suppose that H2 ⊂ H1. We characterize, for subgroups of G, when quasiconvexity relative to H1 implies quasiconvexity relative to H2. We also show that quasiconvexity relative toH2 implies quasiconvexity relative toH1. As an application, we give some ...

To a hyperbolic smooth curve defined over a number-field one naturally associates an "anabelian" representation of the absolute Galois group of the base field landing in outer automorphism group of the algebraic fundamental group. In this paper, we introduce several deformation problems for Lie-algebra versions of the above representation and show that, this way we get a richer structure than t...

We prove that a group is hyperbolic relative to virtually nilpotent subgroups if and only if there exists a Gromov-hyperbolic metric space with bounded geometry on which it acts as a relatively hyperbolic group. As a consequence we obtain that any group hyperbolic relative to virtually nilpotent subgroups has finite asymptotic dimension. For these groups the Novikov conjecture holds. The class ...