The mapping class group orbit of a multicurve

TL;DR

This paper investigates the existence of almost invariant colorings of the mapping class group orbit of multicurves on surfaces, proving their non-existence for genus at least two and their existence for tori.

Contribution

It establishes the non-existence of almost invariant colorings for high-genus surfaces and constructs such colorings for tori, revealing genus-dependent behavior.

Findings

No almost invariant coloring exists for genus ≥ 2 surfaces.

Almost invariant colorings can be constructed for tori.

Colorings can use arbitrarily many colors on tori.

Abstract

Given a set equipped with a transitive action of a group, we define the notion of an almost invariant coloring of the set. We consider the mapping class group orbit of a multicurve on a compact surface, and prove that in the case of genus at least two, no such almost invariant coloring exists. Conversely, in the case of a closed torus, one may find almost invariant colorings using arbitrarily many colors.