Ideally, all infinite type surfaces can be triangulated

TL;DR

This paper proves that all infinite type surfaces can be decomposed into ideal triangles and characterizes which sets of arcs can be extended to such triangulations based on intersection properties.

Contribution

It establishes that every infinite type surface admits an ideal triangulation and provides a criterion for completing arc sets into triangulations.

Findings

All infinite type surfaces admit ideal triangulations.

A set of disjoint arcs can be completed into a triangulation iff it intersects every simple closed curve finitely.

Provides a characterization of triangulable arc sets on infinite surfaces.

Abstract

We show that any surface of infinite type admits an ideal triangulation. Furthermore, we show that a set of disjoint arcs can be completed into a triangulation if and only if, as a set, they intersect every simple closed curve a finite number of times.