Actions, Length Functions, and Non-Archimedean Words

Kharlampovich, Olga; Myasnikov, Alexei; Serbin, Denis

doi:10.1142/s0218196713400031

Cited by 11 publications

(26 citation statements)

References 115 publications

(301 reference statements)

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“…Some of the prominent examples in this class are limit groups in the sense of Sela (see [9,Theorem 0.3]), which coincide with the finitely presented fully residually free groups, as well as groups acting freely on R n -trees (see [19,Theorem 7.1]). More generally, it was shown in [23,Theorem 63] that finitely presented Λ-free groups, where Λ is an ordered abelian group, are also hyperbolic relative to non-cyclic abelian subgroups. Theorem 1.1 also applies to groups that are hyperbolic with respect to virtually abelian groups, since finitely generated virtually abelian groups are (P {1,3,5} )-completable.…”

Section: Introductionmentioning

confidence: 99%

Finite generating sets of relatively hyperbolic groups and applications to geodesic languages

Antolín

Ciobanu

2016

Trans. Amer. Math. Soc.

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Abstract. Given a finitely generated relatively hyperbolic group G, we construct a finite generating set X of G such that (G, X) has the 'falsification by fellow traveler property' provided that the parabolic subgroups {Hω}ω∈Ω have this property with respect to the generating sets {X ∩ Hω}ω∈Ω. This implies that groups hyperbolic relative to virtually abelian subgroups, which include all limit groups and groups acting freely on R n -trees, or geometrically finite hyperbolic groups, have generating sets for which the language of geodesics is regular, and the complete growth series and complete geodesic series are rational. As an application of our techniques, we prove that if each Hω admits a geodesic biautomatic structure over X ∩ Hω, then G has a geodesic biautomatic structure.Similarly, we construct a finite generating set X of G such that (G, X) has the 'bounded conjugacy diagrams' property or the 'neighbouring shorter conjugate' property if the parabolic subgroups {Hω}ω∈Ω have this property with respect to the generating sets {X ∩Hω}ω∈Ω. This implies that a group hyperbolic relative to abelian subgroups has a generating set for which its Cayley graph has bounded conjugacy diagrams, a fact we use to give a cubic time algorithm to solve the conjugacy problem. Another corollary of our results is that groups hyperbolic relative to virtually abelian subgroups have a regular language of conjugacy geodesics.

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Section: Introductionmentioning

confidence: 99%

Finite generating sets of relatively hyperbolic groups and applications to geodesic languages

Antolín

Ciobanu

2016

Trans. Amer. Math. Soc.

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“…Not every torsion-free hyperbolic group is LO ( [8], Theorem 8). Every limit group is a subgroup of a non-standard free group (that is an ultrapower of a free group) and, therefore, is bi-orderable (see for instance in [26]).…”

Section: Groups Satisfying Kaplansky's Unit Conjecturementioning

confidence: 99%

What does a group algebra of a free group “know” about the group?

Kharlampovich

Myasnikov

2018

Annals of Pure and Applied Logic

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“…From the viewpoint of Bass-Serre theory, the question of free actions of finitely generated groups became very important. There is a book [25] on the subject and many new results were obtained in [69]. This is a topic of interest of the first author.…”

Section: Introductionmentioning

confidence: 99%

Beyond Serre's "Trees" in two directions: $Λ$--trees and products of trees

Kharlampovich,

Vdovina

2017

Preprint

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Serre [125] laid down the fundamentals of the theory of groups acting on simplicial trees. In particular, Bass-Serre theory makes it possible to extract information about the structure of a group from its action on a simplicial tree. Serre's original motivation was to understand the structure of certain algebraic groups whose Bruhat-Tits buildings are trees. In this survey we will discuss the following generalizations of ideas from [125]: the theory of isometric group actions on Λ-trees and the theory of lattices in the product of trees where we describe in more detail results on arithmetic groups acting on a product of trees.

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Actions, Length Functions, and Non-Archimedean Words

Cited by 11 publications

References 115 publications

Finite generating sets of relatively hyperbolic groups and applications to geodesic languages

Finite generating sets of relatively hyperbolic groups and applications to geodesic languages

What does a group algebra of a free group “know” about the group?

Beyond Serre's "Trees" in two directions: $Λ$--trees and products of trees

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