Groups, Geometry, and Dynamics


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Volume 5, Issue 2, 2011, pp. 301–325
DOI: 10.4171/GGD/129

Published online: 2011-03-06

On the difficulty of presenting finitely presentable groups

Martin R. Bridson[1] and Henry Wilton[2]

(1) University of Oxford, UK
(2) University of Cambridge, Great Britain

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.

Keywords: Finitely presentable groups, hyperbolic groups, linear groups, decision problems

Bridson Martin, Wilton Henry: On the difficulty of presenting finitely presentable groups. Groups Geom. Dyn. 5 (2011), 301-325. doi: 10.4171/GGD/129