For each d ∈ ℕ, we construct a 3-generated group Hd, which is a subdirect product of free groups, such that the cohomological dimension of Hd is d. Given a group F and a normal subgroup N ⊳ F we prove that any right angled Artin group containing the special HNN-extension of F with respect to N must also contain F/N. We apply this to construct, for every d ∈ ℕ, a 4-generated group Gd, embeddable into a right angled Artin group, such that the cohomological dimension of Gd is 2 but the cohomological dimension of any right angled Artin group, containing Gd, is at least d. These examples are used to show the non-existence of certain "universal" right angled Artin groups. We also investigate finitely presented subgroups of direct products of limit groups. In particular, we show that for every n ∈ ℕ there exists δ(n) ∈ ℕ such that any n-generated finitely presented subgroup of a direct product of finitely many free groups embeds into the δ(n)-th direct power of the free group of rank 2. As another corollary we derive that any n-generated finitely presented residually free group embeds into the direct product of at most δ(n) limit groups.
Non-∀-homogeneity in free groups
