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In this note we partially answer a question posed by Colbois, Dryden, and ElSoufi. Consider the space of constant-volume Riemannian metrics on a connectedmanifold M which are invariant under the action of a discrete Lie group G. Weshow that the first eigenvalue of the Laplacian is not bounded above on thisspace, provided M = S^n, G acts freely, and S^n/G with the round metric admitsa Killing vector field of constant length, or provided M is a compact Lie groupnot equal to T^n, and G is a discrete subgroup of M.

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