Moduli of triangles in the Heisenberg group

We study triangles in the three-dimensional Heisenberg group, $${\mathbb{H}}$$ , equipped with the Carnot-Carathéodory geometry. The set of ordered triangles in $${\mathbb{H}}$$ (excluding certain degenerate triangles), up to congruence is naturally identified via a parametrization map to a fine moduli space of parameters. We determine the homeomorphism type of this moduli space and also that of the coarse moduli space of unordered triangles. We describe a boundary for the fine moduli space and construct a compactification for it, up to similarity under the non-isotropic dilation of $${\mathbb{H}}$$ . Additionally, some trigonometric results for the Carnot-Carathéodory geometry of $${\mathbb{H}}$$ are given: an angle deficit formula and an analog of the Law of Sines in Euclidean geometry.