The language of geodesics for the discrete Heisenberg group

In this paper, we give a complete description of the language of geodesic words for the discrete Heisenberg group $H(\mathbb{Z})$ with respect to the standard two-element generating set. More precisely, we prove that the only dead end elements in $H(\mathbb{Z})$ are nontrivial elements of the commutator subgroup. We give a description of their geodesic representatives, which are called dead end words. The description is based on a minimal perimeter polyomino concept. Finally, we prove that any geodesic word in $H(\mathbb{Z})$ is a prefix of a dead end word.