Infinite groups

Noskov, G. A.; Ремесленников, В. Н.; Roman'kov, V. A.

doi:10.1007/bf01091962

Cited by 15 publications

(13 citation statements)

References 646 publications

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“…There are even examples of non-isomorphic pairs of virtually cyclic groups with isomorphic profinite completions [2]. Nevertheless, many important questions of this type remain open, of which the following question of Remeslennikov is one of the most notable [32,Question 15]. Question 0.2 (Remeslennikov).…”

Section: Corollary Bmentioning

confidence: 99%

Essential surfaces in graph pairs

Wilton¹

2018

J. Amer. Math. Soc.

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A well known question of Gromov asks whether every one-ended hyperbolic group Γ has a surface subgroup. We give a positive answer when Γ is the fundamental group of a graph of free groups with cyclic edge groups. As a result, Gromov's question is reduced (modulo a technical assumption on 2-torsion) to the case when Γ is rigid. We also find surface subgroups in limit groups. It follows that a limit group with the same profinite completion as a free group must in fact be free, which answers a question of Remeslennikov in this case.

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Section: Corollary Bmentioning

confidence: 99%

Essential surfaces in graph pairs

Wilton¹

2018

J. Amer. Math. Soc.

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“…Thus it seems like a natural line of inquiry to try to distinguish the fundamental groups of 3-manifolds by looking at their finite quotients, since finite quotients of fundamental groups correspond to finite-sheeted regular coverings, so that commensurability of 3-manifolds coincides with commensurability of their fundamental groups. Tying together all of the information about finite quotients of fundamental groups through an inverse limit leads us to the concept of the profinite completion of the group (see Definition 2.1), and to profinite rigidity (see Definition 2.6); this thus follows in the path laid down by [12], [14], and [18], among many others. In particular, Wilton and Zaleskii in [17,Thm 8.4] show that the profinite completion of a geometric 3-manifold group determines its geometry, and further, in [18], they show that the profinite completion of a 3-manifold group also determines the JSJ decomposition of the manifold.…”

Section: Introductionmentioning

confidence: 99%

On the profinite distinguishability of hyperbolic Dehn fillings of finite-volume 3-manifolds

Rapoport

2021

Preprint

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“…. , x k ), i. e., the left-hand side of an equation of the form (1). Also, an equation can be written in the form u(x 1 , .…”

Section: Introductionmentioning

confidence: 99%

On Solvability of Regular Equations in the Variety of Metabelian Groups

Roman'kov¹

2017

Applied Discrete Mathematics

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We study the solvability of equations over groups within a given variety or another class of groups. The classes of nilpotent and solvable groups were considered as main classes to investigate from such point of view. The natural analogues of the famous Kervaire Laudenbach and Levin conjectures were raised to the challenge. It was also noted that the "solvable" version of the known theorem by Brodskiǐ is not true. In this paper, for each n ∈ N, n 2, we prove that every regular equation over the free metabelian group M n is solvable in the class M of all metabelian groups. Moreover, there is a metabelian groupM n that contains a solution of every unimodular equation over M n. These results are extended to the class of rigid metabelian groups. Also, we give an example showing that there exists an equation over a locally indicable torsionfree metabelian group G that has no solution in any solvable overgroup of G. It follows that solvable versions of the Levin conjecture are not true. Another example presents an unimodular equation over a locally indicable torsion-free metabelian group G that has no solution in any metabelian overgroup of G. Hence, the Kervaire Laudenbach conjecture is not valid for the variety of all metabelian groups. We prove that there is an unimodular equation over a finite metabelian group G that has no solutions in any finite metabelian overgroup of G. This means that analog of the famous theorem by Gerstenhaber and Rothaus (about solvability of each unimodular equation over a finite group G in some finite overgroup of G) is not valid for the class of finite metabelian groups.

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Infinite groups

Cited by 15 publications

References 646 publications

Essential surfaces in graph pairs

Essential surfaces in graph pairs

On the profinite distinguishability of hyperbolic Dehn fillings of finite-volume 3-manifolds

On Solvability of Regular Equations in the Variety of Metabelian Groups

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