Cubical Geometry in the Polygonalisation Complex

TL;DR

This paper introduces the polygonalisation complex for surfaces, a cube complex model related to the mapping class group, and explores its geometric properties, hyperplanes, and automorphisms, connecting it to the arc complex.

Contribution

It defines the polygonalisation complex, studies its hyperplanes and crossing graph, and characterizes automorphisms, providing new geometric insights into the mapping class group.

Findings

The crossing graph of hyperplanes is quasi-isometric to the arc complex.

Different surfaces have generically different polygonalisation complexes.

The automorphism group of the complex is typically the extended mapping class group.

Abstract

We introduce the polygonalisation complex of a surface, a cube complex whose vertices correspond to polygonalisations. This is a geometric model for the mapping class group and it is motivated by works of Harer, Mosher and Penner. Using properties of the flip graph, we show that the midcubes in the polygonalisation complex can be extended to a family of embedded and separating hyperplanes, parametrised by the arcs in the surface. We study the crossing graph of these hyperplanes and prove that it is quasi-isometric to the arc complex. We use the crossing graph to prove that, generically, different surfaces have different polygonalisation complexes. The polygonalisation complex is not CAT(0), but we can characterise the vertices where Gromov's link condition fails. This gives a tool for proving that, generically, the automorphism group of the polygonalisation complex is the (extended) mapping class group of the surface.