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February 2022 Symmetries of exotic negatively curved manifolds
Mauricio Bustamante, Bena Tshishiku
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J. Differential Geom. 120(2): 231-250 (February 2022). DOI: 10.4310/jdg/1645207478

Abstract

Let $N$ be a smooth manifold that is homeomorphic but not diffeomorphic to a closed hyperbolic manifold $M$. In this paper, we study the extent to which $N$ admits as much symmetry as $M$. Our main results are examples of $N$ that exhibit two extremes of behavior. On the one hand, we find $N$ with maximal symmetry, i.e. $\operatorname{Isom}(M)$ acts on $N$ by isometries with respect to some negatively curved metric on $N$. For these examples, the order of $\operatorname{Isom}(M)$ can be arbitrarily large. On the other hand, we find $N$ with little symmetry, i.e. no subgroup of $\operatorname{Isom}(M)$ of “small” index acts by diffeomorphisms of $N$. The construction of these examples incorporates a variety of techniques including smoothing theory and the Belolipetsky–Lubotzky method for constructing hyperbolic manifolds with a prescribed isometry group.

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Mauricio Bustamante. Bena Tshishiku. "Symmetries of exotic negatively curved manifolds." J. Differential Geom. 120 (2) 231 - 250, February 2022. https://doi.org/10.4310/jdg/1645207478

Information

Received: 7 January 2019; Accepted: 19 September 2019; Published: February 2022
First available in Project Euclid: 23 February 2022

Digital Object Identifier: 10.4310/jdg/1645207478

Subjects:
Primary: 57R55 , 57R55 , 58D19

Keywords: aspherical manifolds , exotic spheres , group actions , negative curvature , Nielsen realization

Rights: Copyright © 2022 Lehigh University

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Vol.120 • No. 2 • February 2022
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