On the Unboundedness of the First Eigenvalue of the Laplacian for G-Invariant Metrics

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In this note we partially answer a question posed by Colbois, Dryden, and El Soufi. Consider the space of constant-volume Riemannian metrics on a connected manifold M which are invariant under the action of a discrete Lie group G. We show that the first eigenvalue of the Laplacian is not bounded above on this space, provided M = Sⁿ, G acts freely, and Sⁿ/G with the round metric admits a Killing vector field of constant length, or provided M is a compact Lie group not equal to Tⁿ, and G is a discrete subgroup of M.