Flip graphs for infinite type surfaces

TL;DR

This paper studies flip graphs of triangulations on infinite type surfaces, revealing their complex structure with uncountably many disconnected components and conditions for relating triangulations via flips.

Contribution

It introduces a new flip graph model allowing simultaneous flips for infinite type surfaces and characterizes when two triangulations are connected.

Findings

Flip graphs have uncountably many connected components.

Conditions for relating triangulations via flips are established.

Flip graphs for infinite surfaces are highly disconnected.

Abstract

We associate to triangulations of infinite type surface a type of flip graph where simultaneous flips are allowed. Our main focus is on understanding exactly when two triangulations can be related by a sequence of flips. A consequence of our results is that flip graphs for infinite type surfaces have uncountably many connected components.