<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>C</mml:mi> <mml:mo>*</mml:mo> </mml:msup></mml:math>-simplicity of HNN extensions and groups acting on trees

Bryder, Rasmus Sylvester; Ivanov, Nikolay A.; Omland, Tron

doi:10.5802/aif.3378

Cited by 4 publications

(4 citation statements)

References 27 publications

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“…One may also want to compare our Theorems 5.20 and 5.17 with results obtained in [13]. In particular, we also have the following examples satisfying Theorem 5.20. .…”

Section: Visual Boundary ∂Xmentioning

confidence: 85%

Boundary actions of CAT(0) spaces and their $C^*$-algebras

Ma¹,

Wang²

2022

Preprint

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In this paper, we study boundary actions of CAT(0) spaces from a point of view of topological dynamics and C * -algebras. First, we investigate the actions of right-angled Coexter groups and right-angled Artin groups with finite defining graphs on the visual boundaries and the Nevo-Sageev boundaries of their natural assigned CAT(0) cube complexes. In particular, we establish (strongly) pure infiniteness results for reduced crossed product C * -algebras of these actions through investigating the corresponding CAT(0) cube complexes and establishing necessary dynamical properties such as minimality, topological freeness and pure infiniteness of the actions. In addition, we study actions of fundamental groups of graphs of groups on the visual boundaries of their Bass-Serre trees. We show that the existence of repeatable paths essentially implies that the action is 2-filling, from which, we also obtain a large class of unital Kirchberg algebras. Furthermore, our result also provides a new method in identifying C * -simple generalized Baumslag-Solitar groups. The examples of groups obtained from our method have n-paradoxical towers in the sense of [26]. This class particularly contains non-degenerated free products, Baumslag-Solitar groups and fundamental groups of n-circles or wedge sums of n-circles.

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“…One may also want to compare our Theorems 5.20 and 5.17 with results obtained in [13]. In particular, we also have the following examples satisfying Theorem 5.20. .…”

Section: Visual Boundary ∂Xmentioning

confidence: 85%

Boundary actions of CAT(0) spaces and their $C^*$-algebras

Ma¹,

Wang²

2022

Preprint

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show abstract

“…The example introduced in [3, Section 5] corresponds to the case Σ −1 ∼ = Σ 1 ∼ = Sym(2). Theorem 2.13 corresponds to [3,Proposition 5.11] in this particular case.…”

Section: Group-theoretic Structurementioning

confidence: 92%

“…It follows from [3,Theorem 4.19] that Λ has the unique trace property if and only if ker Λ has the unique trace property. It also follows from [3,Theorem 4.20] that Λ is C * -simple if and only if K −1 or K 1 is trivial or non-amenable provided Λ is a non-ascending HNN-extension and ker Λ is trivial.…”

Section: Notation Definitions Quasi-kernelsmentioning

confidence: 99%

“…Proof. Lemma 2.15 enables us to apply [3,Theorem 4.19] to conclude that Λ has the unique trace property since ker Λ is trivial. It also enables us to apply [3,Theorem 4.20] to conclude that Λ is C * -simple if and only if K −1 and K 1 are non-amenable, which, by Lemma 2.12, is equivalent to the requirement that some of the groups Σ −1 and Σ 1 is non-amenable.…”

Section: Finally We Provementioning

confidence: 99%

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