Cohomology of convex cocompact groups and invariant distributions on limit sets

TL;DR

This paper studies invariant distributions on limit sets of convex cocompact groups acting on negatively curved symmetric spaces, advancing understanding of their cohomology and related geometric properties.

Contribution

It provides new proofs of Hodge theoretic results, improves bounds on critical exponents, and computes various cohomology groups for hyperbolic manifolds.

Findings

Proved Hodge theoretic results for real hyperbolic manifolds.

Improved bounds for critical exponents in quaternionic and Cayley cases.

Computed L^2-cohomology and higher cohomology groups with hyperfunction coefficients.

Abstract

This paper contains a thorough investigation of invariant distributions supported on limit sets of discrete groups acting convex cocompactly on symmetric spaces of negative curvature. It can be considered as a continuation of math.DG/9810146. Based on this investigation we provide proofs of the Hodge theoretic results for the cohomology of real hyperbolic manifolds announced in math.DG/0009038, improve the bounds for the critical exponents obtained by Corlette for the quaternionic and the Cayley case, compute the L^2-cohomology for the corresponding locally symmetric spaces, prove a version of the Harder-Borel conjecture for real hyperbolic manifolds, and compute higher cohomology groups with coefficients in hyperfunctions supported on the limit set.