Isospectral spherical space forms with non-isomorphic fundamental groups in dimensions 7 and 11: an exhaustive search over the groups of Type I

Abstract

Colantonio and Lauret (arXiv:2602.12939) produced the first pair of isospectral spherical space forms with non-isomorphic fundamental groups, on S¹⁵, and asked for the smallest dimension in which such a pair exists; by results of Ikeda the only open dimensions below 15 are 7 and 11, and their own reductions leave, among fundamental groups of Type I, exactly three cells: Γ₄(m,n,r) acting irreducibly on S⁷, Γ₆(m,n,r) acting irreducibly on S¹¹, and Γ₃(m,n,r) acting on S¹¹ through a sum of two irreducible representations. We search all three cells exhaustively with Ikeda's generating function evaluated in a prime field, which certifies non-isospectrality: there is no isospectral pair with non-isomorphic Type I fundamental groups of order at most 100 000 in dimension 7 (33 682 groups), none of order at most 200 000 in dimension 11 with d=6, and none of order at most 10 000 in dimension 11 with d=3 and two summands (777 groups, every pair of summands) — a cell the source's algorithm, which searches irreducible space forms only, never covered at any order. As controls, the 55 pairs of the source's Table 1 are recomputed by two independent implementations, its S¹⁵ pair is shown almost conjugate by an exact comparison of eigenvalue multisets, and its Proposition 5.2 and Remark 4.5 are re-derived over every parameter tuple of the cell each concerns (d=2, resp. d=4 with n=8) of order at most 100 000. All statements are finite facts about the groups Γ_d(m,n,r), their representations and Ikeda's rational function, verified in Lean 4.

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