Symmetries of exotic negatively curved manifolds
Journal of Differential Geometry
International Press of Boston, Inc.
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Bustamante, M., & Tshishiku, B. Symmetries of exotic negatively curved manifolds. Journal of Differential Geometry https://doi.org/10.17863/CAM.45188
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. Isom(M) acts on N by isometries with respect to some negatively curved metric on N. For these examples, Isom(M) can be made arbitrarily large. On the other hand, we find N with little symmetry, i.e. no subgroup of 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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This record's DOI: https://doi.org/10.17863/CAM.45188
This record's URL: https://www.repository.cam.ac.uk/handle/1810/298131
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