Zero entropy subgroups of mapping class groups

TL;DR

This paper investigates the relationship between the algebraic structure of group actions on surfaces with boundary and their dynamical properties, showing that non-solvable permutation groups imply the existence of pseudo-Anosov components.

Contribution

It establishes a link between the solvability of the induced permutation group and the presence of pseudo-Anosov elements in the mapping class group relative to finite sets.

Findings

Non-solvable permutation groups imply pseudo-Anosov components.

Mapping class groups without pseudo-Anosov elements are solvable in many cases.

Results apply to surfaces with boundary, especially when boundary is non-empty.

Abstract

Let $M$ be a compact surface with boundary. We are interested in the question of how a group action on $M$ permutes a finite invariant set $X \subset int(M)$. More precisely, how the algebraic properties of the induced group of permutations of a finite invariant set affects the dynamical properties of the group. Our main result shows that in many circumstances if the induced permutation group is not solvable then among the homeomorphisms in the group there must be one with a pseudo-Anosov component. We formulate this in terms of the mapping class group relative to the finite set and show the stronger result that in many circumstances (e.g. if $\partial M \ne \emptyset$) this mapping class group is itself solvable if it has no elements with pseudo-Anosov components.