# On the difficulty of presenting finitely presentable groups

### Martin R. Bridson

University of Oxford, UK### Henry Wilton

University of Cambridge, Great Britain

## Abstract

We exhibit classes of groups in which the word problem is uniformly solvable but in which there is no algorithm that can compute finite presentations for finitely presentable subgroups. Direct products of hyperbolic groups, groups of integer matrices, and right-angled Coxeter groups form such classes. We discuss related classes of groups in which there *does* exist an algorithm to compute finite presentations for finitely presentable subgroups. We also construct a finitely presented group that has a polynomial Dehn function but in which there is no algorithm to compute the first Betti number of its finitely presentable subgroups.

## Cite this article

Martin R. Bridson, Henry Wilton, On the difficulty of presenting finitely presentable groups. Groups Geom. Dyn. 5 (2011), no. 2, pp. 301–325

DOI 10.4171/GGD/129