The Poisson boundary of Out(FN)

Horbez, Camille

doi:10.1215/00127094-3166308

Cited by 15 publications

(13 citation statements)

References 46 publications

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“…The outer automorphism group of a non-abelian free group, Out(F n ), acts on a number of distinct Gromov hyperbolic spaces, as shown by Bestvina and Feighn [BF10,BF14] and Handel and Mosher [HM13], and so is weakly hyperbolic. Similarly to the case of Mod(S), convergence to the boundary also follows by considering the action of Out(F n ) on the (locally compact) outer space, as shown by Horbez [Hor14].…”

Section: Examples and Discussionmentioning

confidence: 88%

Random walks on weakly hyperbolic groups

Maher¹,

Tiozzo

2016

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

109

189

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Let G be a countable group which acts by isometries on a separable, but not necessarily proper, Gromov hyperbolic space X. We say the action of G is weakly hyperbolic if G contains two independent hyperbolic isometries. We show that a random walk on such G converges to the Gromov boundary almost surely. We apply the convergence result to show linear progress and linear growth of translation length, without any assumptions on the moments of the random walk.If the action is acylindrical, and the random walk has finite entropy and finite logarithmic moment, we show that the Gromov boundary with the hitting measure is the Poisson boundary.

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Section: Examples and Discussionmentioning

confidence: 88%

Random walks on weakly hyperbolic groups

Maher¹,

Tiozzo

2016

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

109

189

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“…Note that the identification of the Poisson boundary for

Out (F_{n})

has been obtained by Horbez [32] using the action of

Out (F_{n})

on the outer space

C V_{n}

. This gives an identification of the Poisson boundary with both

\partial C V_{n}

and

\partial FF (F_{n})

, as there is a coarsely defined Lipschitz map

C V_{n} \to FF false(F_{n} false)

.…”

Section: Introductionmentioning

confidence: 96%

Random walks, WPD actions, and the Cremona group

Maher

Tiozzo

2021

Proc. London Math. Soc.

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We study random walks on the Cremona group. We show that almost surely the dynamical degree of a sequence of random Cremona transformations grows exponentially fast, and a random walk produces infinitely many different normal subgroups with probability 1. Moreover, we study the structure of such random subgroups. We prove these results in general for groups of isometries of (non‐proper) hyperbolic spaces which possess at least one WPD element. As another application, we answer a question of Margalit showing that a random normal subgroup of the mapping class group is free.

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“…Letμ be the associated exit measure on ∂Γ. Then, it is known by work of [36] and [59] that forμ-almost every z ∈ ∂Γ, there exists a class of R-tree [T z ] ∈ ∂CV N which is free, arational, and uniquely ergodic. The following question is based on this result and the results of [22] and [39].…”

Section: Geometric Structure Of the Cannon-thurston Mapmentioning

confidence: 99%

Trees, dendrites and the Cannon–Thurston map

Field

2020

Algebr. Geom. Topol.

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When 1 → H → G → Q → 1 is a short exact sequence of three word-hyperbolic groups, Mahan Mitra (Mj) has shown that the inclusion map from H to G extends continuously to a map between the Gromov boundaries of H and G. This boundary map is known as the Cannon-Thurston map. In this context, Mitra associates to every point z in the Gromov boundary of Q an "ending lamination" on H which consists of pairs of distinct points in the boundary of H. We prove that for each such z, the quotient of the Gromov boundary of H by the equivalence relation generated by this ending lamination is a dendrite, that is, a tree-like topological space. This result generalizes the work of Kapovich-Lustig and Dowdall-Kapovich-Taylor, who prove that in the case where H is a free group and Q is a convex cocompact purely atoroidal subgroup of Out(F N ), one can identify the resultant quotient space with a certain R-tree in the boundary of Culler-Vogtmann's Outer space.

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The Poisson boundary of Out(FN)

Cited by 15 publications

References 46 publications

Random walks on weakly hyperbolic groups

Random walks on weakly hyperbolic groups

Random walks, WPD actions, and the Cremona group

Trees, dendrites and the Cannon–Thurston map

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