Random Gromov’s monsters do not act non-elementarily on hyperbolic spaces

Gruber, Dominik; Sisto, Alessandro; Tessera, Romain

doi:10.1090/proc/14754

Cited by 10 publications

(10 citation statements)

References 15 publications

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“…Note that in [45] the previous result is proven under the assumption that

X

is separable , that is, it contains a countable dense set. However, since the measure

μ

is countable one can drop the separability assumption, as remarked in [30, Remark 4]. In fact, the only point where separability is used is to prove convergence to the boundary, and one can prove it for general metric spaces from the separable case and the following fact.…”

Section: Background Materialsmentioning

confidence: 99%

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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“…Note that in [45] the previous result is proven under the assumption that

X

is separable , that is, it contains a countable dense set. However, since the measure

μ

Section: Background Materialsmentioning

confidence: 99%

Random walks, WPD actions, and the Cremona group

Maher

Tiozzo

2021

Proc. London Math. Soc.

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“…In our work, we assume that all spaces are separable in order to apply results from . However, Gruber–Sisto–Tessera [, Remark 4] observe that in the current setting one may drop this hypothesis by replacing

X

with a separable metric space quasi‐isometric to it.…”

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confidence: 99%

Counting loxodromics for hyperbolic actions

Gekhtman

Taylor

Tiozzo

2018

Journal of Topology

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Let G X be a non-elementary action by isometries of a hyperbolic group G on a (not necessarily proper) hyperbolic metric space X. We show that the set of elements of G which act as loxodromic isometries of X has density one in the word metric on G. That is, for any finite generating set of G, the proportion of elements in G of word length at most n, which are X-loxodromics, approaches 1 as n → ∞. We also establish several results about the behavior in X of the images of typical geodesic rays in G; for example, we prove that they make linear progress in X and converge to the Gromov boundary ∂X. We discuss various applications, in particular to mapping class groups, Out(FN ), and right-angled Artin groups. and its limit in a suitable compactification (when it exists) is usually called a Patterson-Sullivan (PS) measure. In the second case, for each n one has the convolution measure μ n := μ · · · μ n times , which is precisely the distribution of the nth step of the random walk. In this case, the limit measure in a suitable compactification is the hitting measure for the random walk.The two types of counting, however, need not yield the same notion of genericity; indeed, an underlying theme of much research has been the question: Are Patterson-Sullivan measures also hitting measures for a certain random walk?This is for instance the main question of [64]. As Furstenberg showed [28], for a semisimple Lie group G, the appropriate boundary is B = G/P , where P is a minimal parabolic subgroup. Furthermore, the analogue of the Patterson-Sullivan measure is the unique measure ν on B which is invariant by a maximal compact subgroup. In fact, he proved that for any lattice Λ in G there exists a measure on Λ whose hitting measure is ν. Moreover, for groups acting properly and cocompactly on δ-hyperbolic manifolds, Connell and Muchnik [21] showed that there exists a certain random walk which produces the Patterson-Sullivan measure on the boundary. They also extended this result to arbitrary Gibbs measures in the case the space is CAT(−1) [20].However, for actions of hyperbolic groups G on spaces X, if the harmonic measure and PS measure are in the same measure class, then the metrics on G and on X are quasi-isometric [10]. Interestingly, for hyperbolic groups Gouëzel, Mathéus, and Maucourant [30] recently proved that, when considering a word metric on G, harmonic measures for symmetric random walks of finite support are never in the same measure class as Patterson-Sullivan measures, unless the group is virtually free. One should compare this to Ledrappier's result that for compact hyperbolic surfaces the hitting measure for Brownian motion coincides with the PS measure if and only if the surface has constant curvature [49]. There are many examples of harmonic measures which are singular with respect to the PS-type measure, see for example, [39].In this paper, we will use random walk methods to prove results about genericity with respect to counting in balls in the word metric on G. In particular, we consider the case of G a w...

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“…The first major difference was discovered in [15], where it is shown that random Gromov's monsters cannot act non-elementarily on hyperbolic spaces, while infinitely presented graphical small cancellation groups are acylindrically hyperbolic [14].…”

Section: Introductionmentioning

confidence: 99%

Divergence and quasi-isometry classes of random Gromov’s monsters

Gruber¹,

Sisto²

2021

Math. Proc. Camb. Phil. Soc.

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We show that Gromov’s monsters arising from i.i.d. random labellings of expanders (that we call random Gromov’s monsters) have linear divergence along a subsequence, so that in particular they do not contain Morse quasigeodesics, and they are not quasi-isometric to Gromov’s monsters arising from graphical small cancellation labellings of expanders. Moreover, by further studying the divergence function, we show that there are uncountably many quasi-isometry classes of random Gromov’s monsters.

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Random Gromov’s monsters do not act non-elementarily on hyperbolic spaces

Cited by 10 publications

References 15 publications

Random walks, WPD actions, and the Cremona group

Random walks, WPD actions, and the Cremona group

Counting loxodromics for hyperbolic actions

Divergence and quasi-isometry classes of random Gromov’s monsters

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