  We study the geometry of compact geodesic spaces with trivial first Betti
number admitting large finite groups of isometries. We show that if a finite
group $G$ acts by isometries on a compact geodesic space $X$ whose first Betti
number vanishes, then diam$(X) / $diam$(X / G ) \leq 4 \sqrt{ \vert G \vert }$.
  For a group $G$ and a finite symmetric generating set $S$, $P_k(\Gamma (G,
S))$ denotes the 2-dimensional CW-complex whose 1-skeleton is the Cayley graph
$\Gamma$ of $G$ with respect to $S$ and whose 2-cells are $m$-gons for $0 \leq
m \leq k$, defined by the simple graph loops of length $m$ in $\Gamma$, up to
cyclic permutations.
  Let $G$ be a finite abelian group with $\vert G \vert \geq 3$ and $S$ a
symmetric set of generators for which $P_k(\Gamma (G,S))$ has trivial first
Betti number. We show that the first nontrivial eigenvalue $-\lambda_1$ of the
Laplacian on the Cayley graph satisfies $\lambda_1 \geq 2 - 2 \cos ( 2 \pi / k
) $. We also give an explicit upper bound on the diameter of the Cayley graph
of $G$ with respect to $S$ of the form $O (k^2 \vert S \vert \log \vert G \vert
)$. Related explicit bounds for the Cheeger constant and Kazhdan constant of
the pair $(G,S)$ are also obtained.
