Horizontal and Straight Triangulations on Heisenberg Groups

TL;DR

This paper constructs a specialized triangulation of the Heisenberg group using horizontal and straight simplexes with specific regularity properties, providing explicit examples and extending the triangulation to the entire group.

Contribution

It introduces a novel triangulation method for the Heisenberg group employing horizontal and straight simplexes with regularity properties, extending to the whole group.

Findings

Triangulation of $\

Explicit examples of grid and triangulations provided.

Triangulation extends to the entire Heisenberg group.

Abstract

This paper aims to show that there exists a triangulation of the Heisenberg group $\mathbb{H}^n$ into singular simplexes with regularity properties on both the low-dimensional and high-dimensional layers. For low dimensions, we request our simplexes to be horizontal while, for high dimensions, we define a notion of straight simplexes using exponential and logarithmic maps and we require our simplexes to have high-dimensional straight layers. A triangulation with such simplexes is first constructed on a general polyhedral structure and then extended to the whole Heisenberg group. In this paper we also provide some explicit examples of grid and triangulations.