Indecomposability of equivalence relations generated by word hyperbolic groups

Adams, Scot

doi:10.1016/0040-9383(94)90008-6

Cited by 31 publications

(77 citation statements)

References 9 publications

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“…It is an intriguing unsolved problem to find a geometric characterization for treeable groups. Theorem 1.3 seems close in spirit to results of Adams (1988Adams ( , 1994; however, it appears that there is no direct implication from one result to the other (S. Adams, R. Lyons and B. Weiss, private communications). The referee has suggested that perhaps the techniques of Adams (1988) could be used to give an alternative proof of our main results.…”

Section: Definitions (I)mentioning

confidence: 78%

Nonamenable products are not treeable

Pemantle

Peres

2000

Isr. J. Math.

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Abstract. Let X and Y be infinite graphs, such that the automorphism group of X is nonamenable, and the automorphism group of Y has an infinite orbit. We prove that there is no automorphism-invariant measure on the set of spanning trees in the direct product X × Y . This implies that the minimal spanning forest corresponding to i.i.d. edge-weights in such a product, has infinitely many connected components almost surely. §1. Introduction.There are several natural ways to pick a random spanning tree in a finite graph, notably the uniform spanning tree and the minimal spanning tree (for random edge weights).The limits of these models on infinite graphs sometimes give spanning forests, rather than trees. In this note, we present a large class of graphs where one cannot pick a random subtree in an automorphism-invariant manner.Definition. A subtree Γ = (V Γ , E Γ ) in a graph X = (V X , E X ) is an acyclic connected graph, with V Γ ⊂ V X and E Γ ⊂ E X . If V Γ = V X , then Γ is called a spanning tree of X. By identifying a subtree with the indicator function of its edges, we may view the ensemble of spanning trees in X as a Borel set in the compact metrizable space {0, 1} E X .Theorem 1.1. Let X and Y be infinite, locally finite graphs. Suppose thatis a closed nonamenable subgroup of Aut(X), and H ⊂ Aut(Y ) has an infinite orbit. Then there is no G × H-invariant probability measure on the set of spanning trees of the direct product graph X × Y .Given two graphs X = (V X , E X ) and Y = (V Y , E Y ), the direct product graph X × Y has vertex set V X × V Y ; the vertices (x 1 , y 1 ) and (x 2 , y 2 ) in V X × V Y are taken to be adjacent in X × Y iff either x 1 = x 2 and [y 1 , y 2 ] ∈ E Y , or y 1 = y 2 and [x 1 , x 2 ] ∈ E X .1991 Mathematics Subject Classification. Primary 60B99. Secondary 60D05, 20F32 .

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Section: Definitions (I)mentioning

confidence: 78%

Nonamenable products are not treeable

Pemantle

Peres

2000

Isr. J. Math.

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“…He used this significant property in the process of solving a certain problem concerning equivalence relations generated by hyperbolic groups [2].…”

Section: Chaptermentioning

confidence: 99%

“…Adams [2] investigated invariant Borel maps ϕ : X → M (∂Γ) for a recurrent subrelation S of R, where the word "invariant" means that the map ϕ satisfies…”

Section: Chaptermentioning

confidence: 99%

“…For this purpose, it is helpful to know some results about an essentially free, measure-preserving action of a hyperbolic group on a standard probability space. Following Zimmer's idea in [103], Adams [2] proceeded to study essentially free Borel actions of hyperbolic groups. His method shows that any non-elementary hyperbolic group cannot be measure equivalent to the direct product Γ 1 ×Γ 2 , where both Γ 1 and Γ 2 are infinite and either Γ 1 or Γ 2 has an infinite amenable subgroup (see Theorem 4.68).…”

Section: Introductionmentioning

confidence: 99%

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The mapping class group from the viewpoint of measure equivalence theory

Kida

2008

Memoirs of the AMS

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We obtain some classification result for the mapping class groups of compact orientable surfaces in terms of measure equivalence. In particular, the mapping class groups of different closed surfaces can not be measure equivalent. Moreover, we give various examples of discrete groups which are not measure equivalent to the mapping class groups. In the course of the proof, we investigate amenability in a measurable sense for the actions of the mapping class group on the boundary at infinity of the curve complex and on the Thurston boundary. Using this investigation, we prove that the mapping class group of a compact orientable surface is exact.

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“…If this action is orbit equivalent to a product action of two discrete groups Γ 1 and Γ 2 on S 1 × S 2 , then either S 1 or S 2 is essentially finite. [2]. Let us illustrate the idea of the proof very roughly in case Γ is the fundamental group of a manifold M of negative curvature.…”

Section: Amenability Superrigidity and Other Applicationsmentioning

confidence: 99%

Harmonic analysis in rigidity theory

Spatzier¹

Ergodic Theory and Its Connections With Harmonic Analysis

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Indecomposability of equivalence relations generated by word hyperbolic groups

Cited by 31 publications

References 9 publications

Nonamenable products are not treeable

Nonamenable products are not treeable

The mapping class group from the viewpoint of measure equivalence theory

Harmonic analysis in rigidity theory

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