Train track complex of once-punctured torus and 4-punctured sphere

TL;DR

This paper extends the understanding of the train track complex for certain punctured surfaces, specifically the once-punctured torus and 4-punctured sphere, showing proper group actions in these cases.

Contribution

It proves proper discontinuity and cocompactness of the mapping class group action on the train track complex for the once-punctured torus and 4-punctured sphere, filling a gap in previous results.

Findings

Established proper discontinuous action for the surfaces in question.

Redefined complexes for these specific surfaces.

Extended the applicability of train track complex theory.

Abstract

Consider a compact oriented surface $S$ of genus $g \geq 0$ and $m \geq 0$ punctured. The train track complex of $S$ which is defined by Hamenst\"adt is a 1-complex whose vertices are isotopy classes of complete train tracks on $S$. Hamenst\"adt shows that if $3g-3+m \geq 2$, the mapping class group acts properly discontinuously and cocompactly on the train track complex. We will prove corresponding results for the excluded case, namely when $S$ is a once-punctured torus or a 4-punctured sphere. To work this out, we redefinition of two complexes for these surfaces.