Connectedness of spheres in Cayley graphs

TL;DR

This paper introduces the concept of connection thickness in Cayley graphs, explores its dependence on generating sets, and examines its implications for dead-ends, sphere diameters, and related geometric properties.

Contribution

It defines connection thickness, analyzes its behavior in lamplighter groups, and links it to dead-ends, cut sets, and almost-convexity, providing new insights into Cayley graph geometry.

Findings

Connection thickness is bounded for finitely presented one-ended groups.

In lamplighter groups, connection thickness is linear or logarithmic depending on the generating set.

An example shows finite connection thickness can lead to sphere diameters of order n^2.

Abstract

We introduce the notion of connection thickness of spheres in a Cayley graph, related to dead-ends and their retreat depth. It was well-known that connection thickness is bounded for finitely presented one-ended groups. We compute that for natural generating sets of lamplighter groups on a line or on a tree, connection thickness is linear or logarithmic respectively. We show that it depends strongly on the generating set. We give an example where the metric induced at the (finite) thickness of connection gives diameter of order $n^2$ to the sphere of radius $n$. We also discuss the rarity of dead-ends and the relationships of connection thickness with cut sets in percolation theory and with almost-convexity. Finally, we present a list of open questions about spheres in Cayley graphs.