Normalisers in Limit Groups

Let Γ be a limit group, S ⊂ Γ a non-trivial subgroup, and N the normaliser of S. If H1(S,Q) has finite Q-dimension, then S is finitely generated and either N/S is finite or N is abelian. This result has applications to the study of subdirect products of limit groups. Limit groups (or ω-residually free groups) have received a good deal of attention in recent years, largely due to the work of Z. Sela ([18, 19] et seq.). See for example [1, 3, 7, 10, 16]. O. Kharlampovich, A. Myasnikov ([12, 13] et seq.) and others (see, for example, [9, 11, 14]) have studied limit groups extensively under the more traditional name of fully residually free groups, which appears to have been first introduced by B. Baumslag in [2]. This name reflects the fact that limit groups are precisely those finitely generated groups Γ such that for each finite subset T ⊂ Γ there exists a homomorphism from Γ to a free group that is injective on T . Limit groups are also known as ∃-free groups and ω-residually free groups (see [17]). Examples of limit groups include all finitely generated free or free abelian groups, and the fundamental groups of all closed surfaces of Euler characteristic at most −2. The free product of finitely many limit groups is again a limit group, which leads to further examples. More sophisticated examples are described in some of the articles cited above. The purpose of this note is to contribute some results on the subgroup structure of limit groups. (Their relationship to work of others is discussed briefly in Section 3.) Theorem 1. If Γ is a limit group and H ⊂ Γ is a finitely generated, non-trivial subgroup, then either H has finite index in its normaliser or else the normaliser of H is abelian. We shall use the following result to circumvent the difficulty that a priori one does not know that normalisers in limit groups are finitely generated. Theorem 2. Let Γ be a limit group and S ⊂ Γ a subgroup. If H1(S,Q) has finite Qdimension, then S is finitely generated (and hence is a limit group). Theorems 1 and 2 combine to give the result stated in the abstract. Theorem 1 plays an important role in our work on the subdirect products of limit groups [5, 6]. In the present note, we content ourselves with the following easy consequence of Theorem 1. Theorem 3. Suppose that Γ1, . . . ,Γn are limit groups and let S ⊂ Γ1 × ∙ ∙ ∙ × Γn be an arbitrary subgroup. If Li = Γi ∩ S is non-trivial and finitely generated for i ≤ r, then a subgroup S0 ⊂ S of finite index splits as a direct product S0 = L1 × ∙ ∙ ∙ × Lr × (S0 ∩Gr), 2000 Mathematics Subject Classification. 20F65, 20E08, 20F67.