Division in group rings of surface groups

Abstract

We prove a division algorithm for group rings of high genus surface groups and use it to show that some 2-complexes with surface fundamental groups are standard. The division algorithm works somewhat more generally for groups acting on hyperbolic space H n $\mathbb {H}^n$ with large infimum displacement. We give an application of this to cohomological dimension of 2-relator groups acting on H n $\mathbb {H}^n$ and to handle decompositions of hyperbolic n $n$ -manifolds.