Fibre products, non-positive curvature, and decision problems

Journal article

We give a criterion for fibre products to be finitely presented and use it as the basis of a construction that encodes the pathologies of finite group presentations into pairs of groups P ⊂ G where G is a product of hyperbolic groups and P is a finitely presented subgroup. This enables us to prove that there is a finitely presented subgroup P in a biautomatic group G such that the generalized word problem for P ⊂ G is unsolvable and P has an unsolvable conjugacy problem. An additional construction shows that there exists a compact non-positively curved polyhedron X such that π1X is biautomatic and there is no algorithm to decide isomorphism among the finitely presented subgroups of π1X.

Authors: Baumslag, G; Bridson, M; Miller, C; Short, H

• Publication status: Published
• Journal: COMMENTARII MATHEMATICI HELVETICI
• Volume: 75
• Issue: 3
• Pages: 457-477
• Publication date: 2000-01-01
• DOI: 10.1007/s000140050136
• ISSN: 0010-2571
• Language: English
• Keywords: fibre products; hyperbolic groups; automatic groups; non-positive curvature; decision problems; finitely presented groups