Horizontal and straight triangulation on Heisenberg groups (Q6110400)

The Heisenberg group \({\mathbb{H}}^n\), \(n\geq 1\), is a \((2n+1)\)-dimensional, connected, simply connected and nilpotent Lie group. In addition, it is a Carnot group of step \(2\). Thus the Lie algebra of \({\mathbb{H}}^n\) can be written as a direct sum \({\mathfrak{h}}_1\oplus {\mathfrak{h}}_2\), where \({\mathfrak{h}}_1\) and \({\mathfrak{h}}_2\) are called the horizontal and the vertical layer, respectively. In this paper, the author constructs a triangulation of \({\mathbb{H}}^n\) into singular simplices with certain regularity properties. The author defines a singular \(k\)-simplex \(\Delta^k\to {\mathbb{H}}^n\) to be horizontal, if its image is horizontal in \({\mathbb{H}}^n\), i.e., if the tangent vector fields are horizontal. He also defines the notion of a straight simplex by using exponential and logarithmic maps. He proves that \({\mathbb{H}}^n\) admits a triangulation by singular \(k\)-simplices, where the singular \(k\)-simplices are horizontal piecewise linear maps, for \(0<k\leq n\). For \(n+1\leq k\leq 2n+1\), the singular \(k\)-simplices have straight \(k\)-layers.