Structure and finiteness properties of subdirect products of groups

By Martin R. Bridson, Charles F. Miller. 2007

57 citations recorded by the source corpus.

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 three free or surface groups and to a solution of 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 that there is no algorithm to determine the first homology of a finitely generated subgroup.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8ecb84294b3d23328ce33f53aa591c073f7eddee

Citations in the corpus

• On the finite presentation of subdirect products and the nature of residually free groups (2012, unknown): “From [9] (or [11] in the case of surface groups) we know that a finitely presented full subdirect product S of n limit groups Γi must virtually contain the term γn−1 of the lower central series of the product.”