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Isospectral nearly Kähler manifolds

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Abstract

We give a systematic way to construct almost conjugate pairs of finite subgroups of \(\mathrm {Spin}(2n+1)\) and \({{\mathrm{Pin}}}(n)\) for \(n\in {\mathbb {N}}\) sufficiently large. As a geometric application, we give an infinite family of pairs \(M_1^{d_n}\) and \(M_2^{d_n}\) of nearly Kähler manifolds that are isospectral for the Dirac and Laplace operator with increasing dimensions \(d_n>6\). We provide additionally a computation of the volume of (locally) homogeneous six dimensional nearly Kähler manifolds and investigate the existence of Sunada pairs in this dimension.

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Notes

  1. See Tables 2 and 3 for information on normal subgroups of ADE groups.

  2. We refer to [7] for an exhaustive list of homogeneous nearly Kähler manifolds.

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Acknowledgements

The author wishes to thank M. Larsen and T. Finis for discussions concerning results in [17, 18], as well as A. Adem for pointing out a reference for the group cohomological facts used in the proof of Theorem 2.10. He also wants to thank N. Ginoux for several corrections in the previous versions of this paper and G. Weingart for providing help with the algorithmic computations that yield Example 2.11 and his hospitality during the author’s stay in Cuernavaca.

This work was supported by the Max-Planck-institut für Mathematik in den Naturwissenchaften.

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  1. Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstraße 22, 04103, Leipzig, Germany

    J. J. Vásquez

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Communicated by Vicente Cortés.

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Vásquez, J.J. Isospectral nearly Kähler manifolds. Abh. Math. Semin. Univ. Hambg. 88, 23–50 (2018). https://doi.org/10.1007/s12188-017-0185-2

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