The limit set of non-orientable mapping class groups

TL;DR

This paper investigates the properties of the limit set of non-orientable mapping class groups, providing evidence for and against a conjectural analogy with geometrically finite Fuchsian groups, and explores the structure of these limit sets.

Contribution

It establishes new results on the structure of the limit set of non-orientable mapping class groups, including openness, density, and containment properties, and challenges existing conjectures about convex cores.

Findings

The complement of the limit set is open and dense.

The limit set contains all uniquely ergodic foliations.

The convex core is not quasi-convex.

Abstract

We provide evidence both for and against a conjectural analogy between geometrically finite infinite covolume Fuchsian groups and the mapping class group of compact non-orientable surfaces. In the positive direction, we show the complement of the limit set is open and dense. Moreover, we show that the limit set of the mapping class group contains the set of uniquely ergodic foliations and is contained in the set of all projective measured foliations not containing any one-sided leaves, establishing large parts of a conjecture of Gendulphe. In the negative direction, we show that a conjectured convex core is not even quasi-convex, in contrast with the geometrically finite setting.