Atomic Properties of the Hawaiian Earring Group for HNN Extensions

Nakamura, Jun

doi:10.1080/00927872.2014.939273

Cited by 6 publications

(2 citation statements)

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“…Free (abelian) groups were classically shown to be cm-slender, lcH-slender and n-slender (see [23] and [34]). More recent work has shown that torsion-free word hyperbolic groups, right-angled Artin groups, braid groups and many other groups satisfy various of these slenderness conditions (see [17,20,39,44]). Note that a group which is either n-slender, cm-slender, or lccH-slender must be torsion-free.…”

Section: Problem 1 Does There Exist a Finitely Generated (Countable) Torsion-free Acylindrically Hyperbolic Group Which Does Not Admit Unmentioning

confidence: 99%

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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: Problem 1 Does There Exist a Finitely Generated (Countable) Torsion-free Acylindrically Hyperbolic Group Which Does Not Admit Unmentioning

confidence: 99%

“…Results like Theorems A and B are called atomic properties in the literature (see [25] and [44]). Note that in Theorems A and B we are not requiring the group G to be acylindrically hyperbolic.…”

Section: Introductionmentioning

confidence: 99%