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• W1860294406 abstract "AbstractWe show that, given a finitely generated group G as the coordinategroup of a finite system of equations over a torsion-free hyperbolic groupΓ, there is an algorithm which computes the Grushko decomposition ofeach maximal Γ-limit quotient of G, and then computes the primary JSJdecomposition of each freely indecomposable component. Therefore, wecan effectively find the JSJ decomposition of a freely indecomposable Γ-limit group given as the coordinate group of an irreducible system ofequations over Γ. Additionally, we show that a group is a Γ-limit groupif and only if it is an iterated generalized double over Γ. 1 Introduction Given a group G, another group Lis said to be fully residually G (or dis-criminated by G) if, given any finite subset A ⊂ L, 1 ∈/ A, there exists ahomomorphism φ: L→ G, such that φ(a) 6= 1 for all a∈ A. The study ofgroups discriminated by various classes of groups has been ongoing since theearly twentieth century, with increased activity often occuring as other equiv-alent characterizations of such groups, and applications in different fields, arefound. For example, the development of algebraic geometry over groups allowedfinitely generated groups discriminated by a free group Fto be considered as co-ordinate groups of irreducible systems of equations over F. This helped enablesolutions to Tarski’s problems on the elementary theory of free groups, givenindependently by Kharlampovich-Myasnikov and Sela ([24] and [35]).Throughout, we let Γ be an arbitrary fixed torsion-free hyperbolic group.While some of the theory of fully residually free groups can be generalizedeasily to groups discriminated by Γ, in general groups discriminated by Γ aremore difficult to work with for several reasons. Unlike finitely generated fullyresidually free groups, finitely generated fully residually Γ groups can containfinitely generated subgroups with no finite presentation, so algorithmic resultscan often be more difficult to obtain.In [12], Grigorchuk provided the following topological perspective of finitelygenerated fully residually Ggroups, which also has the benefit of providing1" @default.
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• W1860294406 date "2015-01-13" @default.
• W1860294406 modified "2023-09-22" @default.
• W1860294406 title "Effective JSJ decompositions of maximal -limit quotients" @default.
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