4/29/2004

Igor Mineyev
University of Illinois, Urbana and the Institute for Advanced Study

Gromov hyperbolic groups generalize fundamental groups of closed negatively curved manifolds. Bounded cohomology, due to B. E. Johnson, is defined as the usual (singular or bar-construction) cohomology with the additional boundedness assumption on cochains. I will remind both definitions and will discuss the construction of homological bicombings on hyperbolic graphs. With some extra work this provides a cohomological characterization of hyperbolic groups: a finitely presentable group G is hyperbolic if and only if, in dimension 2, the map from the bounded cohomology of G to the usual cohomology is surjective for all bounded QG-modules as coefficients. This surjectivity also holds for all higher dimensions,  a result used by Connes and Moscovici for a proof of the Novikov conjecture for hyperbolic groups. In the recent joint work with N. Monod and Y. Shalom, we construct  and use ideal bicombings on hyperbolic graphs. An ideal bicombing is a homological version of the classical geodesic flow on hyperbolic manifolds.