Compactifying Exchange Graphs I: Annuli and Tubes

TL;DR

This paper introduces asymptotic triangulations of annuli, demonstrating their mutation properties, and uses them to compactify the exchange graph of annular triangulations, extending to tubes.

Contribution

It defines asymptotic triangulations, shows their mutation behavior, and uses them to compactify the exchange graph of annular triangulations.

Findings

Asymptotic triangulations can be mutated like regular triangulations.

The exchange graph of annular triangulations can be compactified using asymptotic limits.

The approach extends to tubes, broadening the scope of the theory.

Abstract

We introduce the notion of an \emph{asymptotic triangulation} of the annulus. We show that asymptotic triangulations can be mutated as the usual triangulations and describe their exchange graph. Viewing asymptotic triangulations as limits of triangulations under the action of the mapping class group, we compactify the exchange graph of the triangulations of the annulus. The cases of tubes are also considered.