The Sub-Riemannian cut locus of $H$-type groups

TL;DR

This paper characterizes the sub-Riemannian cut locus of $H$-type groups, a class of step-two nilpotent groups, showing it coincides with the group's center by analyzing geodesics.

Contribution

It provides a complete description of sub-Riemannian geodesics in $H$-type groups and identifies the cut locus as the center of these groups.

Findings

The cut locus of $H$-type groups is the group's center.

Geodesics in $H$-type groups are fully characterized.

The cut locus corresponds to points with multiple geodesics.

Abstract

In the present paper we give a proof of the fact that the sub-Riemannian cut locus of a wide class of nilpotent groups of step two, called $H$-type groups, starting from the origin corresponds to the center of the group. We obtain this result by completely describing the sub-Riemannian geodesics in the group, and using these to obtain three disjoint sets of points in the group determined by the number of geodesics joining them to the origin.