Uniform uniform exponential growth of subgroups of the mapping class group
Abstract
Let Mod(S) denote the mapping class group of a compact, orientable surface S. We prove that finitely generated subgroups of Mod(S) which are not virtually abelian have uniform exponential growth with minimal growth rate bounded below by a constant depending only, and necessarily, on S. For the proof, we find in any such subgroup explicit free group generators which are "short" in any word metric. Besides bounding growth, this allows a bound on the return probability of simple random walks.
Keywords
Cite
@article{arxiv.0805.0133,
title = {Uniform uniform exponential growth of subgroups of the mapping class group},
author = {Johanna Mangahas},
journal= {arXiv preprint arXiv:0805.0133},
year = {2009}
}
Comments
12 pgs. V5 benefits from referee comments: clearer proof and added reference for Sec. 4; numerical correction in Lemma 2.5 and where it is applied; typos fixed. V4 adds Sec. 4 (growth bound must depend on surface); V3 adds random walks corollary; V2 states stronger result implicit in original proof and corrects formula for growth rate bound