Geometric and Functional Analysis

, Volume 19, Issue 5, pp 1468–1480

Uniform Uniform Exponential Growth of Subgroups of the Mapping Class Group

Article

DOI: 10.1007/s00039-009-0038-y

Cite this article as:
Mangahas, J. Geom. Funct. Anal. (2010) 19: 1468. doi:10.1007/s00039-009-0038-y

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 and phrases

Mapping class group uniform exponential growth 

2000 Mathematics Subject Classification

57M07 20F65 

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA

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