From local to global conjugacy of subgroups of relatively hyperbolic groups

Bogopolski, Oleg; Bux, Kai‐Uwe

doi:10.1142/s021819671750014x

Cited by 3 publications

(3 citation statements)

References 14 publications

(17 reference statements)

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“…Thus, there exists such a neighborhood V. Then there exists an open neighborhood O ⊆ V of identity 1 H such that O (2) ⊆ V. Then ϕ((H 0 ∩ O) (2) ) ⊆ Ell(G S). Arguing as in the proof of Theorem A starting from (9), we obtain that there exists a relatively open subgroup A ≤ H 0 such that ϕ(A) is finite and A is normal in H 0 . Let A be an open subset of H such that H 0 ∩ A = A.…”

Section: Proof Of a Strong Version Of Theorem Amentioning

confidence: 85%

Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups

Bogopolski

Corson

2021

Math. Ann.

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We describe homomorphisms ϕ : H → G for which G is acylindrically hyperbolic and H is a topological group which is either completely metrizable or locally countably compact Hausdorff. It is shown that, in a certain sense, either the image of ϕ is small or ϕ is almost continuous. We also describe homomorphisms from the Hawaiian earring group to G as above. We prove a more precise result for homomorphisms ϕ : H → Mod( ), where H is as above and Mod( ) is the mapping class group of a connected compact surface . In this case there exists an open normal subgroup V ≤ H such that ϕ(V ) is finite. We also prove the analogous statement for homomorphisms ϕ : H → Out(G), where G is a one-ended hyperbolic group. Some automatic continuity results for relatively hyperbolic groups and fundamental groups of graphs of groups are also deduced. As a by-product, we prove that the Hawaiian earring group is acylindrically hyperbolic, but does not admit any universal acylindrical action on a hyperbolic space.

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Section: Proof Of a Strong Version Of Theorem Amentioning

confidence: 85%

Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups

Bogopolski

Corson

2021

Math. Ann.

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“…Also, O. Bogopolski and K-U. Bux in [5] proved that surface groups are conjugacy subgroup separable.…”

Section: Introductionmentioning

confidence: 98%

“…After this paper was submitted Bogopolski and Bux put the paper [5] into arxiv, where they gave independent prove of Theorems 1.2 and 1.3 in [5, Lemma 6.2 and Corollary C] for torsion free groups. Our methods allow us to avoid the assumption of torsion freeness.…”

Section: Introductionmentioning

confidence: 99%

Limit groups are subgroup conjugacy separable

Chagas

Zalesskiĭ

2016

Journal of Algebra

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A group G is called subgroup conjugacy separable if for every pair of non-conjugate finitely generated subgroups of G, there exists a finite quotient of G where the images of these subgroups are not conjugate. We prove that limit groups are subgroup conjugacy separable. We also prove this property for one relator groups of the form R = a 1 , ..., a n | W n with n > |W |. The property is also proved for virtual retracts (equivalently for quasiconvex subgroups) of hyperbolic virtually special groups.

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Equations in acylindrically hyperbolic groups and verbal closedness

Bogopolski

2022

Groups Geom. Dyn.

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Let H be an acylindrically hyperbolic group without nontrivial finite normal subgroups. We show that any finite system S of equations with constants from H is equivalent to a single equation. We also show that the algebraic set associated with S is, up to conjugacy, a projection of the algebraic set associated with a single splitted equation (such an equation has the form w(x_1,\ldots,x_n)=h , where w\in F(X) , h\in H ). From this we deduce the following statement: Let G be an arbitrary overgroup of the above group H . Then H is verbally closed in G if and only if it is algebraically closed in G . These statements have interesting implications; here we give only two of them: If H is a non-cyclic torsion-free hyperbolic group, then every (possibly infinite) system of equations with finitely many variables and with constants from H is equivalent to a single equation. We give a positive solution to Problem 5.2 from the paper [J. Group Theory 17 (2014), 29–40] of Myasnikov and Roman’kov: Verbally closed subgroups of torsion-free hyperbolic groups are retracts. Moreover, we describe solutions of the equation x^ny^m=a^nb^m in acylindrically hyperbolic groups (AH-groups), where a , b are non-commensurable jointly special loxodromic elements and n,m are integers with sufficiently large common divisor. We also prove the existence of special test words in AH-groups and give an application to endomorphisms of AH-groups.

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From local to global conjugacy of subgroups of relatively hyperbolic groups

Cited by 3 publications

References 14 publications

Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups

Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups

Limit groups are subgroup conjugacy separable

Equations in acylindrically hyperbolic groups and verbal closedness

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