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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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