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.
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