  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.
