Hall’s Theorem for limit groups

A celebrated theorem of Marshall Hall Jr. implies that finitely generated free groups are subgroup separable and that all of their finitely generated subgroups are retracts of finite-index subgroups. We use topological techniques inspired by the work of Stallings to prove that all limit groups share these two properties. This answers a question of Sela. Limit groups are finitely presented groups that arise naturally in many different aspects of the study of finitely generated (non-abelian) free groups. Perhaps their most satisfying characterization is as the closure of the set of free groups in the topology on marked groups that arose from the work of M. Gromov and R. Grigorchuk (see [4]). Limit groups admit a hierarchical decomposition in which the basic building blocks are free groups, free abelian groups and the fundamental groups of surfaces of Euler characteristic less than -1. Therefore, it seems natural to try to generalize properties of these ubiquitous classes of groups to limit groups. Much of the recent work on limit groups has been motivated by the fundamental role that they play in the study of Hom(G,F), the variety of homomorphisms from a finitely generated group G to a free group F, and in the first-order logic of the free group. One can associate to a group G the elementary theory of G, the set of sentences in first-order logic that hold in G. The elementary theory contains the existential theory, which consists of those sentences that use only one, existential, quantifier. From this point of ∗Partially supported by an EPSRC student scholarship and by a post-doctoral fellowship at the Hebrew University of Jerusalem, Israel. 1 ar X iv :m at h/ 06 05 54 6v 4 [ m at h. G R ] 7 J un 2 00 7 view limit groups are precisely the finitely generated groups with the same existential theory as a free group [17]. In [20, 21, 23, 22, 24, 26, 25], Zlil Sela solved a famous problem of Alfred Tarski by classifying the elementarily free groups, those limit groups with the same elementary theory as a free group. O. Kharlampovich and A. Miasnikov also announced a solution to Tarski’s Problem in [11, 12, 14, 13, 15]. Work on Tarski’s Problem has led to the development of a powerful structure theory for limit groups. One form of this is given by a theorem of Kharlampovich and Miasnikov [12] (see also [4]). Let ICE be the smallest set of groups containing all finitely generated free groups that is closed under extending centralizers. Limit groups are precisely the finitely generated subgroups of groups in ICE . We make extensive use of this characterization. Marshall Hall Jr. proved in [7] that every finitely generated subgroup of a finitely generated free group is a free factor in a finite-index subgroup, and that this finite-index subgroup can be chosen to exclude any element not in the original subgroup. As a consequence, finitely generated free groups are subgroup separable (also known as LERF ); that is, every finitely generated subgroup is closed in the profinite topology. This is a strong algebraic condition that implies, for instance, that the generalized word problem is solvable. In the 1970s it became apparent that subgroup separability has a natural topological interpretation. Peter Scott used hyperbolic geometry in [18] to prove that surface groups are subgroup separable, while J. R. Stallings exploited the topology of graphs to reprove Hall’s Theorem and other properties of free groups in [29]. Stallings’ techniques have been extended by, among others, Rita Gitik [6], who proved that, in certain circumstances, the amalgamated product of a subgroup separable group with a free group over a cyclic subgroup is subgroup separable, and D. T. Wise [33], who classified the subgroup separable graphs of free groups with cyclic edge groups. Sela [27] asked if limit groups are subgroup separable. In [30], the author answered Sela’s question in the affirmative for elementarily free groups. Here we extend that result. Theorem A (Corollary 3.9) Limit groups are subgroup separable. It follows that limit groups have solvable generalized word problem. We also prove the stronger theorem that ICE groups are coset separable with respect to vertex groups. Another strong consequence of Hall’s Theorem is that every finitely generated subgroup H of a free group F is a virtual retract ; that is, H is a