Aperiodic tilings of manifolds of intermediate growth

TL;DR

This paper presents a homological method to construct aperiodic tiles on specific open Riemannian surfaces that are acted upon by Grigorchuk groups, which have intermediate growth rates.

Contribution

It introduces a novel homological construction of aperiodic tiles tailored for surfaces with actions of Grigorchuk groups, expanding tiling theory to new geometric contexts.

Findings

Constructed explicit aperiodic tiles for certain Riemannian surfaces.

Demonstrated the applicability of Grigorchuk group actions in tiling theory.

Extended the understanding of aperiodic tilings to manifolds of intermediate growth.

Abstract

We give a homological construction of aperiodic tiles for certain open Riemannian surfaces admitting actions of Grigorchuk groups of intermediate growth.