  We study the monopole Floer homology of a SOLV rational homology sphere Y
from the point of view of spectral theory. Applying ideas of Fourier analysis
on solvable groups, we show that for suitable SOLV metrics on Y, small regular
perturbations of the Seiberg-Witten equations do not admit irreducible
solutions; in particular, this provides a geometric proof that Y is an L-space.
