Here we obtain upper bounds for diameters of covering spaces of compact
length spaces with the lifted metric. We give a proof by Sergei Ivanov of the
fact that the diameter of an $n$-fold cover of a space of diameter $d$ is
bounded above by $dn$. We also show that the bound can be improved to
$4d\sqrt{n}$ if the cover is universal

Zamora, Sergio

English

arXiv

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

On the Diameter of Universal Cover

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