Isospectral spherical space forms and orbifolds of highest volume

TL;DR

The paper resolves the maximal-volume problem for isospectral, non-isometric spherical orbifolds and space forms in dimension $d\ge5$ by leveraging spectral generating functions $F_{\Gamma}(z)$ obtained from Molien's formula to extract group-theoretic invariants. It proves that the largest possible orbifold volume is $\mathrm{vol}(S^{d})/8$, realized by almost-conjugate subgroups of order $8$, and shows that lens-space cases with small group orders are spectrally rigid up to isometry, yielding exact maximal-volume bounds in many dimensions. For odd-dimensional space forms, the paper proves $\mathrm{vol}(S^{2n-1})/11$ is maximal in infinite families dictated by congruence conditions, using detailed analysis of the $\varphi(q)$-restricted lens-space spectra and their principal parts. It also corrects previous claims about non-cyclic fundamental groups, showing a richer landscape of isospectral-but-distinct families parameterized by a discrete variable $h$, with spectra distinguished by specific $F^{(k)}(z)$ invariants, thereby refining the classification of isospectral spherical space forms.

Abstract

We prove that $\operatorname{vol}(S^{d})/8$ is the highest volume of a pair of $d$-dimensional isospectral and non-isometric spherical orbifolds for any $d\geq5$. Furthermore, we show that $\operatorname{vol}(S^{2n-1})/11$ is the highest volume of a pair of $(2n-1)$-dimensional isospectral and non-isometric spherical space forms if either $n\geq11$ and $n\equiv 1\pmod 5$, or $n\geq7$ and $n\equiv 2\pmod 5$, or $n\geq3$ and $n\equiv 3\pmod 5$.

Theorems & Definitions (19)

• Theorem 1.2
• Theorem 1.4
• Theorem 1.5
• Remark 2.1
• Remark 2.2
• Proposition 2.3
• Proposition 2.4
• Lemma 2.6
• proof
• Remark 2.7