Metric Properties of Diestel-Leader Groups

TL;DR

This paper explores the metric properties of Diestel-Leader groups, revealing complex geometric features like dead end elements and diverse cone types, using a combinatorial approach to analyze their Cayley graphs.

Contribution

It introduces a combinatorial formula for word length in Diestel-Leader groups and demonstrates their intricate metric structure, including dead end elements and infinite cone types.

Findings

Existence of dead end elements of arbitrary depth

Infinitely many cone types in these groups

No regular language of geodesics

Abstract

In this paper we investigate metric properties of the groups $\Gamma_d(q)$ whose Cayley graphs are the Diestel-Leader graphs $DL_d(q)$ with respect to a given generating set $S_{d,q}$. These groups provide a geometric generalization of the family of lamplighter groups, whose Cayley graphs with respect to a certain generating set are the Diestel-Leader graphs $DL_2(q)$. Bartholdi, Neuhauser and Woess in \cite{BNW} show that for $d \geq 3$, $\Gamma_d(q)$ is of type $F_{d-1}$ but not $F_d$. We show below that these groups have dead end elements of arbitrary depth with respect to the generating set $S_{d,q}$, as well as infinitely many cone types and hence no regular language of geodesics. These results are proven using a combinatorial formula to compute the word length of group elements with respect to $S_{d,q}$ which is also proven in the paper and relies on the geometry of the Diestel-Leader graphs.