The language of geodesics for the discrete Heisenberg group
Abstract
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 twoelement 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 arXiv:
 arXiv:1905.03226
 Bibcode:
 2019arXiv190503226A
 Keywords:

 Mathematics  Group Theory