Right-angled hexagon tilings of the hyperbolic plane

TL;DR

This paper investigates invariant probability measures on hyperbolic plane tilings with right-angled hexagons, establishing the existence of measures with prescribed tile distributions.

Contribution

It proves the existence of isometry-invariant measures on hyperbolic tilings with given tile shape distributions, extending understanding of hyperbolic tiling measures.

Findings

Existence of invariant measures with prescribed tile distributions

Construction of measures for a natural family of right-angled hexagon shapes

Extension of measure theory to hyperbolic tilings

Abstract

We study isometry-invariant probability measures on the space $\Omega$ of tilings of the hyperbolic plane with right-angled hexagons of varying shapes. We prove that, for each measure $\mu$ in a certain natural family of measures on right-angled hexagons, there is an isometry-invariant measure on $\Omega$ whose marginal distribution on tiles is $\mu$.