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Joseph Maher

Publications and source records attributed to Joseph Maher.

At least 19 recordsLinked to original sources
arXiv2026

Singularity of Cannon-Thurston maps

In a closed fibered hyperbolic 3-manifold M ,the inclusion of a fiber S, with S and M lifted to the universal covers, gives an exponentially distorted embedding of the hyperbolic plane into hyperbolic 3-space. Nevertheless, Cannon and Thurston showed that there is a map from the circle at infinity of the hyperbolic plane to the 2-sphere at infinity of hyperbolic 3-space. The Cannon-Thurston map is surjective, finite-to-one, and gives a space-filling curve. Here we use properties of geodesics to prove that many natural measures on the circle when pushed forward by the Cannon-Thurston map become singular with respect to many natural measures on the 2-sphere. The circle measures we consider are the Lebesgue measure and stationary measures that arise from fully supported random walks on the surface group. The measures on the sphere we consider are the Lebesgue measure and stationary measures that arise from geometric random walks on the 3- manifold group. We obtain the singularity of measures from the following properties of typical geodesics. We prove that a hyperbolic geodesic sampled with respect to a pushforward measure asymptotically spends a definite proportion of its time close to a fiber. On the other hand, we show that a hyperbolic geodesic sampled with respect to a natural measure on the sphere spends an asymptotically negligible proportion of its time close to a fiber. For a more restricted class of circle measures, namely the Lebesgue measure and stationary measures from geometric random walks on the surface group, we also prove an effective result for the proportion of time spent close to a fiber.

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arXiv2025

Quasi-geodesics in the Cannon-Thurston metric

A closed fibered 3-manifold admits a complete hyperbolic metric if and only if it has a fibration with a pseudo-Anosov monodromy. The stable and the unstable laminations associated to the pseudo-Anosov homeomorphism on the fiber surface give rise to a natural metric on the 3-manifold, the Cannon-Thurston metric, which is quasi-isometric to the hyperbolic metric. In this paper, we describe a specific family of quasi-geodesics in the Cannon-Thurston metric. We use the main results of this article in a companion paper to obtain statistics for typical geodesics with respect to various natural measures on the 2-sphere, thus giving a geometric criterion for singularity between some of these measure classes.

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arXiv2021

Random trees in the boundary of Outer space

We prove that for the harmonic measure associated to a random walk on Out$(F_r)$ satisfying some mild conditions, a typical tree in the boundary of Outer space is trivalent and nongeometric. This answers a question of M. Bestvina.

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arXiv2021

Random walks, WPD actions, and the Cremona group

We study random walks on groups of isometries of non-proper delta-hyperbolic spaces under the assumption that at least one element in the group satisfies Bestvina-Fujiwara's WPD condition. We show that in this case typical elements are WPD, and the Poisson boundary coincides with the Gromov boundary. Moreover, we show that the random walk satisfies a form of asymptotic acylindricality, and we use this to show that the normal closure of random elements yields almost surely infinitely many different normal subgroups. Moreover, the probability that the normal closure is free tends to 1 if and only if the maximal normal subgroup coincides with the center of the group. We apply such techniques to the Cremona group, thus obtaining that the dynamical degree of random Cremona transformations grows exponentially fast, producing many different normal subgroups, and identifying the Poisson boundary. We also give a new identification of the Poisson boundary of Out(F_n). Our methods give bounds on the rates of convergence for these results.

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arXiv2020

Quotients of the curve complex

We consider three kinds of quotients of the curve complex which are obtained by coning off uniformly quasi-convex subspaces: symmetric curve sets, non-maximal train track sets, and compression body disc sets. We show that the actions of the mapping class group on those quotients are strongly WPD, which implies that the actions are non-elementary and those quotients are of infinite diameter.

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arXiv2017

Morse functions to graphs and topological complexity for hyperbolic 3-manifolds

Scharlemann and Thompson define the width of a 3-manifold M as a notion of complexity based on the topology of M. Their original definition had the property that the adjacency relation on handles gave a linear order on handles, but here we consider a more general definition due to Saito, Scharlemann and Schultens, in which the adjacency relation on handles may give an arbitrary graph. We show that for compact hyperbolic 3-manifolds, this is linearly related to a notion of metric complexity, based on the areas of level sets of Morse functions to graphs, which we call Gromov area.

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arXiv2017

Recurrence of quadratic differentials for harmonic measure

We consider random walks on the mapping class group that have finite first moment with respect to the word metric, whose support generates a non-elementary subgroup and contains a pseudo-Anosov map whose invariant Teichmuller geodesic is in the principal stratum of quadratic differentials. We show that a Teichmuller geodesic typical with respect to the harmonic measure for such random walks, is recurrent to the thick part of the principal stratum. As a consequence, the vertical and horizontal foliations of such a random Teichmuller geodesic have no saddle connections.

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arXiv2017

Random subgroups of acylindrically hyperbolic groups and hyperbolic embeddings

Let G be an acylindrically hyperbolic group. We consider a random subgroup H in G, generated by a finite collection of independent random walks. We show that, with asymptotic probability one, such a random subgroup H of G is a free group, and the semidirect product of H acting on E(G) is hyperbolically embedded in G, where E(G) is the unique maximal finite normal subgroup of G.

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arXiv2016

Exponential decay in the mapping class group

We show that the probability that a finitely supported random walk on a non-elementary subgroup of the the mapping class group gives a non-pseudo-Anosov element decays exponentially in the length of the random walk. More generally, we show that if R is a set of mapping class group elements with an upper bound on their translation lengths on the complex of curves, then the probability that a random walk lies in R decays exponentially in the length of the random walk.

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arXiv2016

The stratum of random mapping classes

We consider random walks on the mapping class group whose support generates a non-elementary 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.

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arXiv2015

Morse area and Scharlemann-Thompson width for hyperbolic 3-manifolds

Scharlemann and Thompson define a numerical complexity for a 3-manifold using handle decompositions of the manifold. We show that for compact hyperbolic 3-manifolds this is linearly related to a definition of metric complexity in terms of the areas of level sets of Morse functions.

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arXiv2015

Random walks on weakly hyperbolic groups

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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arXiv2014

Word length statistics for Teichmüller geodesics and singularity of harmonic measure

Given a measure on the Thurston boundary of Teichmueller 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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arXiv2014

Random methods in 3-manifold theory

We show that for any integers k and g, with g at least two, there are infinitely many closed hyperbolic 3-manifolds which are integral homology spheres with Casson invariant k, and Heegaard genus equal to g. This existence result is shown using random methods, using a model of random 3-manifolds arising from random walks on the mapping class group of a closed orientable surface.

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arXiv2013

Statistics and compression of scl

We obtain sharp estimates on the growth rate of stable commutator length on random (geodesic) words, and on random walks, in hyperbolic groups and groups acting nondegenerately on hyperbolic spaces. In either case, we show that with high probability stable commutator length of an element of length $n$ is of order $n/\log{n}$. This establishes quantitative refinements of qualitative results of Bestvina-Fujiwara and others on the infinite dimensionality of 2-dimensional bounded cohomology in groups acting suitably on hyperbolic spaces, in the sense that we can control the geometry of the unit balls in these normed vector spaces (or rather, in random subspaces of their normed duals). As a corollary of our methods, we show that an element obtained by random walk of length $n$ in a mapping class group cannot be written as a product of fewer than $O(n/\log{n})$ reducible elements, with probability going to 1 as $n$ goes to infinity. We also show that the translation length on the complex of free factors of a random walk of length $n$ on the outer automorphism group of a free group grows linearly in $n$.

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arXiv2010

Random walks on the mapping class group

We show that a random walk on the mapping class group of an orientable surface gives rise to a pseudo-Anosov element with asymptotic probability one. Our methods apply to many subgroups of the mapping class group, including the Torelli group.

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