Eigenvalue multiplicity and volume growth
  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 $G$. 
James R. Lee and Yury Makarychev