The language of geodesics for the discrete Heisenberg group

TL;DR

This paper characterizes the structure of geodesic words in the discrete Heisenberg group, identifying dead end elements and their geodesic representatives, and showing all geodesics are prefixes of dead end words.

Contribution

It provides a complete description of geodesic words in the discrete Heisenberg group, including dead end elements and their minimal perimeter polyomino representations.

Findings

Dead end elements are exactly the nontrivial elements of the commutator subgroup.

Geodesic words are prefixes of dead end words.

Dead end words are characterized by minimal perimeter polyominoes.

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 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.