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.
1
University of Oxford, UK
2
University of Cambridge, Great Britain
Martin R. Bridson; Henry Wilton. On the difficulty of presenting finitely presentable groups. Groups, geometry, and dynamics, Tome 5 (2011) no. 2, pp. 301-325. doi: 10.4171/ggd/129
@article{10_4171_ggd_129,
author = {Martin R. Bridson and Henry Wilton},
title = {On the difficulty of presenting finitely presentable groups},
journal = {Groups, geometry, and dynamics},
pages = {301--325},
year = {2011},
volume = {5},
number = {2},
doi = {10.4171/ggd/129},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/129/}
}
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AU - Martin R. Bridson
AU - Henry Wilton
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