Groups, Geometry, and Dynamics
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On the difficulty of presenting finitely presentable groups
Martin R. Bridson (1) and Henry Wilton (2)
(1) Mathematical Institute, University of Oxford, 24-29 St Giles', OX1 3LB, OXFORD, UNITED KINGDOM(2) Department of Mathematics, University College London, WC1E 6BT, LONDON, 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