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Random walks on the mapping class group. (English) Zbl 1213.37072
The mapping class group of an orientable surface of finite type (a surface of genus \(g\) with \(p\) marked points called punctures) consists of orientation-preserving mappings that preserve the punctures modulo those mappings isotopic to the identity. Thurston’s work shows that all elements of the mapping class group are periodic, reducible or pseudo-Anosov. The author of this paper proves that a random walk on the mapping class group reaches a pseudo-Anosov element with asymptotic probability one.
In fact, the author proves something broader. His main theorem is as follows: On the mapping class group \(G\) of an orientable non-sporadic surface of finite type, consider a random walk determined by a probability distribution \(\mu\) whose support generates a nonelementary subgroup. If \(R\) is a subset of \(G\) with the property that every element of \(R\) is conjugate to an element of relative length at most \(B\) for some constant \(B\), then the probability that a random walk of length \(n\) lies in \(R\) tends to zero as \(n\to\infty\).

MSC:
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
60G50 Sums of independent random variables; random walks
20F65 Geometric group theory
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