Homological aperiodic tilings of 3-dimensional geometries

TL;DR

This paper constructs the first known aperiodic tiles for certain 3-dimensional Lie groups, using higher-dimensional homology, thereby resolving the existence question for all non-compact 3-manifold geometries.

Contribution

It introduces a novel method for creating aperiodic tiles in 3D Lie groups using higher-dimensional uniformly finite homology, completing the classification for non-compact geometries.

Findings

First aperiodic tiles for Sol and Heisenberg groups

Complete classification for non-compact 3-manifold geometries

New application of higher-dimensional homology in tiling theory

Abstract

We construct the first aperiodic tiles for two amenable 3-dimensional Lie groups: Sol and the Heisenberg group. Our construction relies on the use of higher-dimensional uniformly finite homology. In particular, we settle completely the existence of aperiodic tiles for all of the non-compact geometries of 3-manifolds appearing in the geometrization conjecture.