On the ideal triangulation graph of a punctured surface

TL;DR

This paper investigates the structure of the ideal triangulation graph of a punctured surface, establishing its automorphism group correspondence with the extended mapping class group and analyzing its Gromov hyperbolicity properties.

Contribution

It proves that for most punctured surfaces, the automorphism group of the ideal triangulation graph matches the extended mapping class group, and shows that this graph is not Gromov hyperbolic.

Findings

Automorphism group of $T(S)$ is isomorphic to the extended mapping class group for most surfaces.

The ideal triangulation graph $T(S)$ is not Gromov hyperbolic under the studied conditions.

Distinct from the curve complex, $T(S)$ exhibits different hyperbolic properties.

Abstract

We study the ideal triangulation graph $T(S)$ of a punctured surface $S$ of finite type. We show that if $S$ is not the sphere with at most three punctures or the torus with one puncture, then the natural map from the extended mapping class group of $S$ into the simplicial automorphism group of $T(S)$ is an isomorphism. We also show that under the same conditions on $S$, the graph $T(S)$ equipped with its natural simplicial metric is not Gromov hyperbolic. Thus, from the point of view of Gromov hyperbolicity, the situation of $T(S)$ is different from that of the curve complex of $S$.