Orbifold lens spaces that are isospectral but not isometric
Two Riemannian manifolds are said to be isospectral if they have the same eigenvalues of the Laplace operator. The notion of isospectrality can be extended to Riemannian orbifolds, in which coordinate charts are modeled on quotients of Riemannian manifolds by finite groups of isometries; a standard inverse spectral problem in this setting is whether there exist isospectral non-isometric orbifolds. Examples of such orbifolds of various flavors have been constructed, including arbitrarily large finite families of isospectral orbifolds [\textit{N. Shams, E. Stanhope} and \textit{D. L. Webb}, Arch. Math. 87, No.~4, 375--385 (2006; Zbl 1111.58029)] and isospectral orbifolds with maximal isotropy groups of different orders [\textit{J. P. Rossetti, D. Schueth} and \textit{M. Weilandt}, Ann. Global Anal. Geom. 34, No.~4, 351--366 (2008; Zbl 1166.58016)]. In [Ann. Sci. Éc. Norm. Supér. (4) 13, 303--315 (1980; Zbl 0451.58037)], \textit{A. Ikeda} constructed isospectral non-isometric lens spaces; the paper under review extends this work to the orbifold setting. An orbifold lens space is a quotient of the round sphere by a cyclic group of orthogonal transformations, where the group is permitted to have fixed points. The author shows that, for every dimension greater than 8, there exist pairs of isospectral non-isometric orbifold lens spaces. The proof relies on giving a convenient formulation of the generating functions of orbifold lens spaces and showing that, under certain conditions, there are sufficiently many generating functions to guarantee the existence of isospectral non-isometric orbifold lens spaces.
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