The Heisenberg plane

Trettel, Steve

doi:10.2140/agt.2023.23.1463

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The Heisenberg plane

Steve Trettel

Algebraic & Geometric Topology 23 (2023) 1463–1500

DOI: 10.2140/agt.2023.23.1463

Abstract

The geometry of the Heisenberg group acting on the plane arises naturally in geometric topology as a degeneration of the familiar spaces $𝕊^{2}, ℍ^{2}$ and $𝔼^{2}$ 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.

Keywords

low-dimensional, geometric topology, geometric transition, cone manifold, geometric structure, Heisenberg group

Mathematical Subject Classification 2010

Primary: 57M50

References

Publication

Received: 10 May 2018

Revised: 5 October 2021

Accepted: 14 November 2021

Published: 14 June 2023

Authors

Steve Trettel
Department of Mathematics Stanford University Stanford, CA United States
http://www.stevejtrettel.site

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Volume 23, issue 4 (2023)