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Limit groups for relatively hyperbolic groups, I: The basic tools
"... Abstract. We begin the investigation of Γlimit groups, where Γ is a torsionfree group which is hyperbolic relative to a collection of free abelian subgroups. Using the results of [16], we adapt the results from [22]. Specifically, given a finitely generated group G, and a sequence of pairwise non ..."
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Cited by 14 (5 self)
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Abstract. We begin the investigation of Γlimit groups, where Γ is a torsionfree group which is hyperbolic relative to a collection of free abelian subgroups. Using the results of [16], we adapt the results from [22]. Specifically, given a finitely generated group G, and a sequence of pairwise nonconjugate homomorphisms {hn: G → Γ}, we extract an Rtree with a nontrivial isometric Gaction. We then provide an analogue of Sela’s shortening argument. In his remarkable series of papers [40, 42], Z. Sela has classified those finitely generated groups with the same elementary theory as the free group of rank 2 (see also [41] 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 longstanding questions of Tarski (Kharlampovich and Miasnikov have another approach to these problems; see [30]). In [40], 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 fullyresidually free group. He then produces MakaninRazborov diagrams, which give a parametrization of Hom(G, F), where G is an arbitrary finitely generated group and F is a nonabelian free group (such a parametrisation is also given in [29]). Over the course of his six papers, two of the main tools Sela uses are the theory of isometric actions on Rtrees and the shortening argument. Sela’s work naturally raises the question of which other classes of groups can be understood using this approach. Many of Sela’s methods (and, more strikingly, some of the answers) come from geometric group theory. Thus, when looking for classes of groups to apply these methods to, it seems natural to consider groups of interest in geometric group theory. In [43], Sela considers an arbitrary torsionfree hyperbolic group Γ, and characterises those groups with the same elementary theory as Γ. Of particular note is his results that any group which has
Algebraic logic, varieties of algebras, and algebraic varieties
, 1995
"... Abstract. The aim of the paper is discussion of connections between the three kinds of objects named in the title. In a sense, it is a survey of such connections; however, some new directions are also considered. This relates, especially, to sections 3, 4 and 5, where we consider a field that could ..."
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Cited by 13 (5 self)
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Abstract. The aim of the paper is discussion of connections between the three kinds of objects named in the title. In a sense, it is a survey of such connections; however, some new directions are also considered. This relates, especially, to sections 3, 4 and 5, where we consider a field that could be understood as an universal algebraic geometry. This geometry is parallel to universal algebra. In the monograph [51] algebraic logic was used for building up a model of a database. Later on, the structures arising there turned out to be useful for solving several problems from algebra. This is the position which the present paper is written from.