The Structure of Limit Groups over Hyperbolic Groups

Let Γ be a torsion-free hyperbolic group. We study Γ–limit groups which, unlike the fundamental case in which Γ is free, may not be finitely presentable or geometrically tractable. We define model Γ–limit groups, which always have good geometric properties (in particular, they are always relatively hyperbolic). Given a strict resolution of an arbitrary Γ–limit group L, we canonically construct a strict resolution of a model Γ–limit group, which encodes all homomorphisms L → Γ that factor through the given resolution. We propose this as the correct framework in which to study Γ–limit groups algorithmically. We enumerate all Γ–limit groups in this framework. Limit groups over a group Γ (otherwise known as Γ–limit groups) arise naturally when studying algebraic geometry over Γ; that is to say, sets of homomorphisms Hom(G,Γ), where G is a finitely generated group. We will be concerned with the case in which Γ is a non-elementary, torsion-free hyperbolic group, a case which has been studied extensively by Sela [45] and others [32, 37, 38, 20]. The results of [45] extend Sela’s solution to Tarski’s problems in the case when Γ is free in [44] et seq. (see also [31] et seq.). Since every finitely generated subgroup of Γ is a Γ–limit group, the hyperbolic case immediately presents new problems that do not arise in the free case. One such problem is that hyperbolic groups typically contain subgroups that are finitely generated but not finitely presented [39]. Thus, it is not immediately clear how to give a finite description of a Γ–limit group. Moreover, Γ–limit groups do not in general have the nice geometric properties of limit groups over free groups (which are all toral relatively hyperbolic [1, 8], in particular finitely presented). The geometry of hyperbolic and relatively hyperbolic groups has been integral to the solutions of many algorithmic problems [40, 42, 10, 12, Date: March 23, 2016. The work of the first author was supported by the National Science Foundation and by a grant from the Simons Foundation (#342049 to Daniel Groves). The second author was supported by the EPSRC..