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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Limits of Geometries


Authors: Daryl Cooper, Jeffrey Danciger and Anna Wienhard
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 57M50; Secondary 22E15, 57S25, 57S20
DOI: https://doi.org/10.1090/tran/7174
Published electronically: March 21, 2018
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Abstract: A geometric transition is a continuous path of geometric structures that changes type, meaning that the model geometry, i.e., the homogeneous space on which the structures are modeled, abruptly changes. In order to rigorously study transitions, one must define a notion of geometric limit at the level of homogeneous spaces, describing the basic process by which one homogeneous geometry may transform into another. We develop a general framework to describe transitions in the context that both geometries involved are represented as sub-geometries of a larger ambient geometry. Specializing to the setting of real projective geometry, we classify the geometric limits of any sub-geometry whose structure group is a symmetric subgroup of the projective general linear group. As an application, we classify all limits of three-dimensional hyperbolic geometry inside of projective geometry, finding Euclidean, Nil, and Sol geometry among the limits. We prove, however, that the other Thurston geometries, in particular $ \mathbb{H}^2 \times \mathbb{R}$ and $ \widetilde {\textup {SL}_2\mathbb{R}}$, do not embed in any limit of hyperbolic geometry in this sense.


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Additional Information

Daryl Cooper
Affiliation: Department of Mathematics, University of California Santa Barbara, South Hall, Room 6607, Santa Barbara, CA 93106-3080
Email: cooper@math.ucsb.edu

Jeffrey Danciger
Affiliation: Department of Mathematics, University of Texas Austin, 1 University Station C1200, Austin, Texas 78712-1202
Email: jdanciger@math.utexas.edu

Anna Wienhard
Affiliation: Ruprecht-Karls Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany — and — Heidelberg Institute for Theoretical Studies, Schloss-Wolfsbrunnenweg 35, 69118 Heidelberg, Germany.
Email: wienhard@uni-heidelberg.de

DOI: https://doi.org/10.1090/tran/7174
Received by editor(s): February 13, 2015
Received by editor(s) in revised form: December 20, 2016
Published electronically: March 21, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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