Random walks on weakly hyperbolic groups

Maher, Joseph F.; Tiozzo, Giulio

doi:10.48550/arxiv.1410.4173

Cited by 16 publications

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“…We can now finish the proof of Theorem A, using the following result of Maher and Tiozzo: Theorem 4.2 (Maher-Tiozzo [MT14]). Let Γ be a countable group of isometries of a separable Gromov hyperbolic space X, let ν be a non-elementary probability distribution on Γ, and let x 0 ∈ X a basepoint.…”

Section: Random Walks On Hyperbolic Spacesmentioning

confidence: 91%

Hyperbolic rigidity of higher rank lattices

Haettel

2016

Preprint

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We prove that any action of a higher rank lattice on a Gromov-hyperbolic space is elementary. More precisely, it is either elliptic or parabolic. This is a large generalization of the fact that any action of a higher rank lattice on a tree has a fixed point. A consequence is that any quasi-action of a higher rank lattice on a tree is elliptic, i.e. it has Manning's property (QFA). Moreover, we obtain a new proof of the theorem of Farb-Kaimanovich-Masur that any morphism from a higher rank lattice to a mapping class group has finite image, without relying on the Margulis normal subgroup theorem nor on bounded cohomology. More generally, we prove that any morphism from a higher rank lattice to a hierarchically hyperbolic group has finite image. In the Appendix, Vincent Guirardel and Camille Horbez deduce rigidity results for morphisms from a higher rank lattice to various outer automorphism groups.

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Section: Random Walks On Hyperbolic Spacesmentioning

confidence: 91%

Hyperbolic rigidity of higher rank lattices

Haettel

2016

Preprint

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“…Theorem 4.3 (Maher [25], Maher-Tiozzo [26]). Let µ be a probability distribution on the mapping class group which has finite first moment in the word metric, and such that the semigroup generated by its support is a non-elementary subgroup.…”

Section: Random Walksmentioning

confidence: 99%

“…Note that in [25], the result is proven under the additional condition that the support of µ is bounded in the relative metric, while such condition is not needed in [26].…”

Section: Random Walksmentioning

confidence: 99%

“…Since G is nonamenable, a random walk makes linear progress in the word metric as shown by Kesten and Day (see Theorem 4.2). Moreover, the random walk makes linear progress in the relative metric, too: Proposition 5.7 (Maher-Tiozzo [26]). Let µ be a probability distribution on a non-compact finite covolume Fuchsian group G which has finite first moment in the word metric, and such that the semigroup generated by its support is a non-elementary subgroup of G. Then there is a constant c > 0 such that…”

Section: Random Walksmentioning

confidence: 99%

“…The result in [26] is stated in general for random walks on (not necessarily proper) Gromov hyperbolic spaces, and it applies here since it is well-known that the Fuchsian group G with the relative metric is δhyperbolic. An earlier result, under the additional hypothesis of convergence to the boundary and finite support, is proven in [25].…”

Section: Random Walksmentioning

confidence: 99%

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Word length statistics for Teichmüller geodesics and singularity of harmonic measure

Gadre¹,

Maher²,

Tiozzo³

2017

Comment. Math. Helv.

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Given a measure on the Thurston boundary of Teichmüller space, one can pick a geodesic ray joining some basepoint to a randomly chosen point on the boundary. Different choices of measures may yield typical geodesics with different geometric properties. In particular, we consider two families of measures: the ones which belong to the Lebesgue or visual measure class, and harmonic measures for random walks on the mapping class group generated by a distribution with finite first moment in the word metric. We consider the ratio between the word metric and the relative metric of approximating mapping class group elements along a geodesic ray, and prove that this ratio tends to infinity along almost all geodesics with respect to Lebesgue measure, while the limit is finite along almost all geodesics with respect to harmonic measure. As a corollary, we establish singularity of harmonic measure. We show the same result for cofinite volume Fuchsian groups with cusps. As an application, we answer a question of Deroin-Kleptsyn-Navas about the vanishing of the Lyapunov expansion exponent.

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The stratum of random mapping classes

Gadre¹,

Maher

2017

Ergod. Th. Dynam. Sys.

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We consider random walks on the mapping class group whose support generates a nonelementary subgroup and contains a pseudo-Anosov map whose invariant Teichmüller geodesic is in the principal stratum. For such random walks, we show that mapping classes along almost every infinite sample path are eventually pseudo-Anosov, with invariant Teichmüller geodesics in the principal stratum. This provides an answer to a question of Kapovich and Pfaff [KP15].We will refer to condition (3) above as the principal stratum assumption.

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Random walks on weakly hyperbolic groups

Cited by 16 publications

References 0 publications

Hyperbolic rigidity of higher rank lattices

Hyperbolic rigidity of higher rank lattices

Word length statistics for Teichmüller geodesics and singularity of harmonic measure

The stratum of random mapping classes

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