Fibre products, non-positive curvature, and decision problems

Abstract

Abstract. 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 π 1 X is biautomatic and there is no algorithm to decide isomorphism among the finitely presented subgroups of π 1 X.

On the difficulty of presenting finitely presentable groups

“…There is a basic template for finding such pathologies: one begins with a complicated finitely presented group, applies some form of the Rips construction to it, and then takes a fibre product (see Subsections 3.3 and 3.4). This scheme originates in [2] and relies on the 1-2-3 Theorem proved there; see also [3]. This paper is organised as follows.…”

“…(1) Q has a universal central extension z Q (and, conversely, the existence of a universal central extension implies that Q is perfect); (2) there is a short exact sequence…”

Non-positive curvature and complexity for finitely presented groups

“…Thus one can encode all of the complexity of finite group-presentations (the lions of figure 1) into the finitely generated subgroups of such H. But such constructions say nothing about finitely presented subgroups because, by a theorem of R. Bieri [14], N is not finitely presentable if Q is infinite. The following theorem from [6] obviates this difficulty.…”

“…Several other applications are given in [6], one of which was refined in [40] to prove that there exist 2-dimensional hyperbolic groups Γ such that there is no algorithm to decide isomorphism among the finitely presented subgroups of Γ × Γ × Γ.…”

“…One applies this criterion to the output of an algorithm obtained by combining D. Wise's refinement [94] of the Rips construction with the 1-2-3 Theorem [6]:…”