Joseph Maher
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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We show that the largest subsurface projection distance between a marking and its image under the nth step of a random walk grows logarithmically in n, with probability approaching 1 as n tends to infinity. Our setup is general and also applies to (relatively) hyperbolic groups and to $\mathrm{Out}(F_n)$. We then use this result to prove Rivin's conjecture that for a random walk $(w_n)$ on the mapping class group, the shortest geodesic in the hyperbolic mapping torus $M_{w_n}$ has length on the order of $1/ \log^2(n)$.
On random walks on the mapping class group
本文通过定义有限型曲面曲线图的电化，将映射类群上随机游走的Poisson边界识别为极小与极大测地叶状结构空间上的平稳测度。
We consider a random walk on the mapping class group of a surface of finite type. We assume that the random walk is determined by a probability measure whose support is finite and generates a non-elementary subgroup $H$. We further assume that $H$ is not consisting only of lifts with respect to any one covering. Then we prove that the probability that such a random walk gives a non-minimal mapping class in its fibered commensurability class decays exponentially. As an application of the minimality, we prove that for the case where a surface has at least one puncture, the probability that a ran…
For any pseudo-Anosov diffeomorphism on a closed orientable surface $S$ of genus greater than one, it is known by the work of Bers and Thurston that the topological entropy agrees with the translation distance on the Teichmüller space with respect to the Teichmüller metric. In this paper, we consider random walks on the mapping class group of $S$. The drift of a random walk is defined as the translation distance of the random walk. We define the topological entropy of a random walk and prove that it almost surely agrees with the drift on the Teichmüller space with respect to the Teichmüller me…
We establish spectral theorems for random walks on mapping class groups of connected, closed, oriented, hyperbolic surfaces, and on $\text{Out}(F_N)$. In both cases, we relate the asymptotics of the stretching factor of the diffeomorphism/automorphism obtained at time $n$ of the random walk to the Lyapunov exponent of the walk, which gives the typical growth rate of the length of a curve -- or of a conjugacy class in $F_N$ -- under a random product of diffeomorphisms/automorphisms. In the mapping class group case, we first observe that the drift of the random walk in the curve complex is also …
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.
This survey is concerned with random walks on mapping class groups. We illustrate how the actions of mapping class groups on Teichmüller spaces or curve complexes reveal the nature of random walks, and vice versa. Our emphasis is on the analogues of classical theorems, including laws of large numbers and central limit theorems, and the properties of harmonic measures, under optimal moment conditions. We also explain the geometric analogy between Gromov hyperbolic spaces and Teichmüller spaces that has been used to copy the properties of random walks from one to the other.
A random walk on a countable group $G$ acting on a metric space $X$ gives a characteristic called the drift which depends only on the transition probability measure $μ$ of the random walk. The drift is the `translation distance' of the random walk. In this paper, we prove that the drift varies continuously with the transition probability measures, under the assumption that the distance and the horofunctions on $X$ are expressed by certain ratios. As an example, we consider the mapping class group $\mathrm{MCG}(S)$ acting on the Teichmüller space. By using north-south dynamics, we also consider…
We show that simple random walks on (non-trivial) relatively hyperbolic groups stay $O(\log(n))$-close to geodesics, where $n$ is the number of steps of the walk. Using similar techniques we show that simple random walks in mapping class groups stay $O(\sqrt{n\log(n)})$-close to geodesics and hierarchy paths. Along the way, we also prove a refinement of the result that mapping class groups have quadratic divergence. An application of our theorem for relatively hyperbolic groups is that random triangles in non-trivial relatively hyperbolic groups are $O(\log(n))$-thin, random points have $O(\lo…
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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