Uniform uniform exponential growth of subgroups of the mapping class group

TL;DR

This paper proves that non-virtually abelian finitely generated subgroups of the mapping class group exhibit uniform exponential growth, with a lower bound depending only on the surface, and provides explicit generators and bounds on random walk return probabilities.

Contribution

It establishes uniform exponential growth for these subgroups with explicit generators and bounds, advancing understanding of their algebraic and probabilistic properties.

Findings

Non-virtually abelian subgroups have uniform exponential growth.

Explicit free group generators are found within these subgroups.

Bounds on random walk return probabilities are derived.

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.