Amenable hyperbolic groups

Caprace, Pierre–Emmanuel; Cornulier, Yves de; Monod, Nicolas; Tessera, Romain

doi:10.4171/jems/575

Cited by 68 publications

(194 citation statements)

References 33 publications

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“…The goal of this section is to prove Theorem 1.4 (2). Our approach is to first show that all quasi-parabolic structures of the lamplighter groups are regular, and derive a characterization of quasi-parabolic structures on these groups using techniques developed in [2].…”

Section: Quasi-parabolic Structures On the Lamplighter Groupsmentioning

confidence: 99%

“…The proof of the above theorem is a combination of the description of quasi-parabolic structures obtained (in a different language) in [2], along with elementary (but lengthy) arguments from commutative algebra. Also note that in general, the above equality does not hold.…”

Section: Introductionmentioning

confidence: 99%

“…The following notions are taken from [2]. These will serve as important tools in proving Theorem 1.4.…”

Section: Introductionmentioning

confidence: 99%

“…Definition 3.1. [2,Section 4] Let H be a group and Q be a subset of H, and let α be an automorphism of H. We say that the action of α is (strictly) confining H into Q if it satisfies the following conditions :…”

Section: Introductionmentioning

confidence: 99%

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Hyperbolic structures on wreath products

Balasubramanya

2019

Journal of Group Theory

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The study of the poset of hyperbolic structures on a group G was initiated in [1]. However, this poset is still very far from being understood and several questions remain unanswered. In this paper, we give a complete description of the poset of hyperbolic structures on the lamplighter groups Z n wr Z and obtain some partial results about more general wreath products. As a consequence of this result, we answer two open questions regarding quasi-parabolic structures from [1]: we give an example of a group G with an uncountable chain of quasi-parabolic structures and prove that the Lamplighter groups Z n wr Z all have finitely many quasi-parabolic structures.Out of these, lineal and general-type actions were well-examined in [1], and several interesting examples and results were obtained. Of special interest were the following results: given any n ∈ N, there exist (distinct) finitely generated groups G n and H n such that |H (G n )| = n and |H gt (H n )| = n.However, the understanding of quasi-parabolic structures is far from being complete. It was shown in [1, Proposition 4.27] that Z wr Z has an uncountable antichain of quasi-parabolic structures, but little else is known. The authors of [1] consequently posed the following two open questions. Problem 1.1. Does there exist a group G such that H qp (G) is non-empty and finite? Problem 1.2. Does there exist a group G such that H qp (G) contains an uncountable chain?

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Section: Quasi-parabolic Structures On the Lamplighter Groupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The following notions are taken from [2]. These will serve as important tools in proving Theorem 1.4.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hyperbolic structures on wreath products

Balasubramanya

2019

Journal of Group Theory

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“…We can combine the above two examples to get what was coined a millefeuille space in [CdCMT12]. See also [Dym13] for more details on its geometry.…”

Section: The Groups and Their Geometrymentioning

confidence: 99%

Envelopes of certain solvable groups

Dymarz¹

2015

Comment. Math. Helv.

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A discrete subgroup Γ of a locally compact group H is called a uniform lattice if the quotient H/Γ is compact. Such an H is called an envelope of Γ. In this paper we study the problem of classifying envelopes of various solvable groups including the solvable Baumslag-Solitar groups, lamplighter groups and certain abelian-by-cyclic groups. Our techniques are geometric and quasi-isometric in nature. In particular we show that for every Γ we consider there is a finite family of preferred model spaces X such that, up to compact groups, H is a cocompact subgroup of Isom(X). We also answer problem 10.4 in [FM00] for a large class of abelian-by-cyclic groups.

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Percolation on Hyperbolic Graphs

Hutchcroft

2019

Geom. Funct. Anal.

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We prove that Bernoulli bond percolation on any nonamenable, Gromov hyperbolic, quasitransitive graph has a phase in which there are infinitely many infinite clusters, verifying a well-known conjecture of Benjamini and Schramm (1996) under the additional assumption of hyperbolicity. In other words, we show that p c < p u for any such graph. Our proof also yields that the triangle condition ∇ pc < ∞ holds at criticality on any such graph, which is known to imply that several critical exponents exist and take their mean-field values. This gives the first family of examples of one-ended groups all of whose Cayley graphs are proven to have mean-field critical exponents for percolation.

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Amenable hyperbolic groups

Cited by 68 publications

References 33 publications

Hyperbolic structures on wreath products

Hyperbolic structures on wreath products

Envelopes of certain solvable groups

Percolation on Hyperbolic Graphs

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