811 research outputs found

### Random Heegaard splittings

A random Heegaard splitting is a 3-manifold obtained by using a random walk
of length n on the mapping class group as the gluing map between two
handlebodies. We show that the joint distribution of random walks of length n
and their inverses is asymptotically independent, and converges to the product
of the harmonic and reflected harmonic measures defined by the random walk. We
use this to show that the translation length on the curve complex of a random
walk grows linearly in the length of the walk, and similarly, that distance in
the curve complex between the disc sets of a random Heegaard splitting grows
linearly in n. In particular, this implies that a random Heegaard splitting is
hyperbolic with asymptotic probability one.Comment: 31 pages, 5 figures, revised versio

### Period three actions on lens spaces

We show that a free period three action on a lens space is standard, i.e. the
quotient is homeomorphic to a lens space. This is an extension of the result
for period three actions on the three-sphere, arXiv:math.GT/0204077, by the
author and J. Hyam Rubinstein.Comment: 67 pages, 54 picture

### Period three actions on the three-sphere

We show that a free period three action on the three-sphere is standard, i.e.
the quotient is homeomorphic to a lens space. We use a minimax argument
involving sweepouts.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol7/paper11.abs.htm

### Word length statistics for Teichmuller geodesics and singularity of harmonic measure

Given a measure on the Thurston boundary of Teichmuller 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 word length of approximating mapping class group elements along a geodesic ray, and prove that this quantity grows superlinearly in time along almost all geodesics with respect to Lebesgue measure, while along almost all geodesics with respect to harmonic measure the growth is linear. As a corollary, the harmonic and Lebesgue measures are mutually singular. We also prove a similar result for the ratio between the word metric and the relative metric (i.e. the induced metric on the curve complex)

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