Subgroups of direct products of limit groups

Abstract

If 1 ; : : : ; n are limit groups and S 1 n is of type FP n ‫/ޑ.‬ then S contains a subgroup of finite index that is itself a direct product of at most n limit groups. This answers a question of Sela.

On subdirect products of type FP m of limit groups

“…In a sequel of papers [8], [9], [10], Bridson, Howie, Miller and Short have generalized the previously known results to the case of a finitely generated subdirect product of limit groups, i.e. a subgroup of a direct product of limit groups that maps surjectively to every factor.…”

“…In [9] it was shown that if S is a subgroup of the direct product G of n non-abelian limit groups (not necessary di¤erent) such that S intersects every direct factor non-trivially and projects surjectively to every direct summand then either S has finite index in G or there is a subgroup S 0 of finite index in S such that for some j c n the homology group H j ðS 0 ; QÞ is infinite-dimensional. Furthermore some necessary conditions for H j ðS 1 ; QÞ to be finite-dimensional for every subgroup of finite index S 1 in S and for all j c 2 are given in [9,Theorem 4.2]. It was recently shown by the same authors that these conditions are su‰cient and imply that S is finitely presented, hence of type FP 2 over Q; see [10].…”

“…We proceed by induction on the height of the limit group. By [9] a freely indecomposable limit group of height h d 1 is the fundamental group of a finite graph of groups with infinite cyclic edge groups and has at least one vertex G v 0 that is a non-abelian limit group of height at most h À 1. In this case the action of G on the Bass-Serre tree G implies immediately that…”

Non-positive curvature and complexity for finitely presented groups

“…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.…”

A subgroup of a direct product of free groups whose Dehn function has a cubic lower bound

“…They proved that if G is a finitely presented subgroup of F ð1Þ Â F ð2Þ , where F ð1Þ and F ð2Þ are free groups, then G is itself virtually a direct product of at most two free groups. This work was extended by Bridson, Howie, Miller and Short [5] to an arbitrary number of factors. They proved that if G is a subgroup of a direct product of n free groups and if G enjoys the finiteness property FP n , then G is virtually a direct product of at most n free groups.…”