Structure and finiteness properties of subdirect products of groups

Abstract

Abstract. We investigate the structure of subdirect products of groups, particularly their finiteness properties. We pay special attention to the subdirect products of free groups, surface groups and HNN extensions. We prove that a finitely presented subdirect product of free and surface groups virtually contains a term of the lower central series of the direct product or else fails to intersect one of the direct summands. This leads to a characterization of the finitely presented subgroups of the direct product of 3 free or surface groups, and to a solution to the conjugacy problem for arbitrary finitely presented subgroups of direct products of surface groups. We obtain a formula for the first homology of a subdirect product of two free groups and use it to show there is no algorithm to determine the first homology of a finitely generated subgroup.

Non-positive curvature and complexity for finitely presented groups

“…Motivated by this, one would like to understand all finitely presented subdirect products of surface groups. In [41] Miller and I proved the following theorem and a weaker version (involving nilpotent quotients) for products of arbitrarily many surfaces.…”

Subgroups of direct products of limit groups

“…We first prove a general lemma (from [12]) about a subdirect product S of n arbitrary (not necessarily limit) groups 1 ; : : : ; n . As before, we write L i for the normal subgroup S \ i of i .…”

“…The present article represents the culmination of a project to prove that it can. Building on ideas and results from [10], [7], [8], [9], [12] we prove: THEOREM A. If 1 ; : : : ; n are limit groups and S 1 n is a subgroup of type FP n ‫,/ޑ.‬ then S is virtually a direct product of n or fewer limit groups.…”

“…The turn of events that leads us in this direction is explained in Section 4 -it begins with a simple observation about higher commutators from [12] and proceeds via a spectral sequence argument.…”

“…These results suggest that there is a real prospect of understanding all such subgroups, that is, all finitely presented residually free groups. We take up this challenge in [11], where we describe precisely which subdirect products of limit groups are finitely presented and present a solution to the conjugacy and membership problems for these subgroups (see also [12], [13]). But the isomorphism problem for finitely presented residually free groups remains open, and beyond that lie many further challenges.…”