Cayley graphs with an infinite Heesch number

TL;DR

This paper constructs a specific 2-generated group whose Cayley graph contains finite connected subsets with arbitrarily large finite Heesch numbers, advancing understanding of geometric properties of groups.

Contribution

It introduces a novel example of a group with Cayley graph exhibiting finite subsets with unbounded finite Heesch numbers, a new phenomenon in geometric group theory.

Findings

Existence of a 2-generated group with Cayley graph having arbitrarily large finite Heesch numbers.

Finite connected subsets with unbounded finite Heesch numbers in a Cayley graph.

New insights into the geometric structure of groups and their Cayley graphs.

Abstract

We construct a 2-generated group $\Gamma $ such that its Cayley graph possesses finite connected subsets with arbitrarily big finite Heesch number.