Limit groups for relatively hyperbolic groups. I. The basic tools

By D. Groves. 2004

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We begin the investigation of -limit groups, where is a torsion-free group which is hyperbolic relative to a collection of free abelian subgroups. Using the results of (16), we adapt the re- sults from (21) and (22) to this context. Specifically, given a finitely generated group G, and a sequence of pairwise non-conjugate ho- momorphisms {hn : G ! }, we extract an R-tree with a nontrivial isometric G-action. This, along with the analogue of Sela's shortening argument allows us to prove the main result of this paper, that is Hopfian. In his remarkable series of papers (38, 39, 41), Z. Sela has classified those finitely generated groups with the same elementary theory as the free group of rank 2 (see also (40) for a summary). This class includes all nonabelian free groups, most surface groups, and certain other hyperbolic groups. In particular, Sela answers in the positive some long-standing questions of Tarski (Kharlampovich and Miasnikov have another approach to these problems; see (29)). In (38), Sela begins with a study of limit groups. Sela's definition of a limit group is geometric, though it turns out that a group is a limit group if and only if it is a finitely generated fully-residually free group. Sela then produces Makanin-Razborov diagrams, which give a parametrization of Hom(G,F), where G is an arbitrary finitely gener- ated group and F is a nonabelian free group (such a parametrisation is also given in (28)). Over the course of his six papers, two of the main tools Sela uses are the theory of isometric actions on R-trees and the shortening argument. Sela's work naturally raises the question of which other classes of groups can be understood using Sela's approach. Many of Sela's meth- ods (and, more strikingly, some of the answers) come from geometric group theory. Thus it seems natural to consider, when looking for classes of groups to apply these methods to, groups of interest in geo- metric group theory. In (42), Sela considers an arbitrary torsion-free

Subfields: Geometry and Topology, Mathematical Physics

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Citations in the corpus

Krull dimension of solvable groups (2008, background): “Notice, that the limit groups of free groups (known also as fully residually free groups, or freely discriminated groups, or ω-residually free groups) played a crucial part in a recent solution of the Tarski problems [35, 54], now they are one of the most actively studied classes of groups in geomet”
ON THE STRUCTURAL THEORY OF II1 FACTORS OF NEGATIVELY CURVED GROUPS, II. ACTIONS BY PRODUCT GROUPS (2011, unknown): “k(z⊗1)PF(ζn)k ≤ |F| zk2. Fix nand denote by P the orthogonal projection onto L2(M2)⊗¯L2(M¯), i.e. P= P∅. We have PF(ζn) = P h∈F P(ζn(u ∗ h ⊗ 1))(uh ⊗ 1) and by the Cauchy-Schwarz inequality we deduce (15) k(z⊗1)PF(ζn)k2 6|F| X h∈F k(z⊗1)P(ζn(u∗ h ⊗1))k 2 Now we let EM2 to be the conditional expectat”
Unification theorems in algebraic geometry (2008, background): “The limit groups of a fixed group G appear in many different situations: in combinatorial group theory as groups discriminated by G (ωresidually G-groups or fully residually G-groups) [2, 3, 34, 5, 6], in the algebraic geometry over groups as the coordinate groups of irreducible varieties over G [4,”