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The Heisenberg plane

Steve Trettel

Algebraic & Geometric Topology 23 (2023) 1463–1500
Abstract

The geometry of the Heisenberg group acting on the plane arises naturally in geometric topology as a degeneration of the familiar spaces 𝕊2, 2 and 𝔼2 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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