On the difficulty of presenting finitely presentable groups

Details On the difficulty of presenting finitely presentable groups On the difficulty of presenting finitely presentable groups

2010-03-26

arxiv.org/abs/1003.5117v2

Mathematics

Group Theory

Final version. To appear in GGD volume dedicated to Fritz Grunewald

Scientific paper

10.4171/GGD/129

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 the finitely presentable subgroups.

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