  Let $G$ be a finite group with symmetric generating set $S$, and let $c =
\max_{R > 0} |B(2R)|/|B(R)|$ be the doubling constant of the corresponding
Cayley graph, where $B(R)$ denotes an $R$-ball in the word-metric with respect
to $S$. We show that the multiplicity of the $k$th eigenvalue of the Laplacian
on the Cayley graph of $G$ is bounded by a function of only $c$ and $k$. More
specifically, the multiplicity is at most $\exp((\log c)(\log c + \log k))$.
  Similarly, if $X$ is a compact, $n$-dimensional Riemannian manifold with
non-negative Ricci curvature, then the multiplicity of the $k$th eigenvalue of
the Laplace-Beltrami operator on $X$ is at most $\exp(n^2 + n log k)$.
  The first result (for $k=2$) yields the following group-theoretic
application. There exists a normal subgroup $N$ of $G$, with $[G : N] \leq
\alpha(c)$, and such that $N$ admits a homomorphism onto the cyclic group
$Z_M$, where $M \geq |G|^{\delta(c)}$ and $\alpha(c), \delta(c) > 0$ are
explicit functions depending only on $c$. This is the finitary analog of a
theorem of Gromov which states that every infinite group of polynomial growth
has a subgroup of finite index which admits a homomorphism onto the integers.
  This addresses a question of Trevisan, and is proved by scaling down
Kleiner's proof of Gromov's theorem. In particular, we replace the space of
harmonic functions of fixed polynomial growth by the second eigenspace of the
Laplacian on the Cayley graph of $G$.
