The Heisenberg Plane

TL;DR

This paper studies the geometry of the Heisenberg group acting on the plane, classifies orbifolds with Heisenberg structures, and analyzes how these structures deform and regenerate, especially in relation to cone tori limits.

Contribution

It provides a classification of orbifolds with Heisenberg structures and computes their deformation spaces, including a detailed example of regeneration from cone tori.

Findings

Classified all orbifolds admitting Heisenberg structures.

Computed deformation spaces of these orbifolds.

Determined which Heisenberg tori arise as limits of collapsing cone tori.

Abstract

The geometry of the Heisenberg group acting on the plane arises naturally in geometric topology as a degeneration of the familiar spaces $\mathbb{S}^2,\mathbb{H}^2$ and $\mathbb{E}^2$ via conjugacy limit as defined by Cooper, Danciger, and Wienhard. This paper considers the deformation and regeneration of Heisenberg structures on orbifolds, adding a carefully worked low-dimensional example to the existing literature on geometric transitions. In particular, the closed orbifolds admitting Heisenberg structures are classified, and their deformation spaces are computed. Considering the regeneration problem, which Heisenberg tori arise as rescaled limits of collapsing paths of constant curvature cone tori is completely determined in the case of a single cone point