Joseph Maher

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.

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 non-degenerately on hyperbolic spaces. In either case, we show that with high probability stable commutator length of an element of length

Exponential decay in the mapping class group

n

Random Heegaard splittings

is of order

Linear progress in the complex of curves

n/ \log n

Random methods in 3-manifold theory

. This establishes quantitative refinements of qualitative results of Bestvina and Fujiwara and others on the infinite dimensionality of two-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

Period three actions on the three-sphere

n

Virtually embedded boundary slopes

in a mapping class group cannot be written as a product of fewer than

Heegaard gradient and virtual fibers

O(n/ \log n)

Word length statistics for Teichmuller geodesics and singularity of harmonic measure

reducible elements, with probability going to