TITLE: Diameter of universal cover
QUESTION [15 upvotes]: Let $M$ be Riemannian manifold and $\tilde M$ be its universal cover (with induced metric).
What is the upper bound for $k=\mathop{diam}\tilde M/\mathop{diam} M$ in terms of $m=|\pi_1(M)|$ (or $\pi_1(M)$)?
Comments:

There is a similar answered question here, but there  cover is NOT universal. So we get $k\leqslant m$, but $k\ll m$ is expected.
The question is open even in case $\pi_1=\mathbb Z_m$ (even asymptotics is not known).   
Clearly, $\sup k$ for given finite group $\Gamma=\pi_1(M)$ is an invariant of $\Gamma$. Is it an interesting invariant?

Examples:

For $\pi_1=\mathbb Z_{3\cdot 2^n}$ one can make $k\sim n$ or $k=O(\log m)$  (see my answer below).
For $\pi_1=S_n$, one can make $k$ of order $n^2$ or (see Greg's answer and also this question). It is much more than $\log m$, but still $k=o(\log^2 m)$.

REPLY [5 votes]: As both Anton and Greg pointed out, it is enough to look at the Cayley graph of a presented finite group $G$ with only quadratic and cubic relators and study how large $diam(G)$ can be in terms of $\vert G \vert$. Note that the property that all relators are quadratic or cubic implies that $G$ is the $1$-skeleton of a simply connected $2$-dimensional simplicial complex. 
It can be shown that $diam(G) \leq \sqrt{ 4 \vert G \vert +1}-2$, which implies the effective bound $diam(\tilde{M} )/ diam(M) \leq 4 \sqrt{\vert \pi_1(M) \vert } $. For the strictly asymptotic behavior, the main result of this paper implies that $diam (G) = o (\vert G \vert ^p) $ for any $p>0$, implying that $diam(\tilde{M} ) / diam(M) = o ( \vert \pi_1 (M) \vert ^p)$ for any $p>0$.
An sketch of the proof of the first inequality follows. Take $h \in G$ with $d(h,e)= diam(G)$ and a path $e=g_0, g_1 , \ldots, g_{diam(G)}=h$. Set $S_i$ to be the induced subgraph by the set $ \{  g \in G  \mid d(g,e)= i \}$ for each $i \in \mathbb{N}$ and $T_i  $ the connected component of $g_i$ in $S_i$. Fix $1 \leq i < diam(G)$. Removing $T_i$ disconnects $G$, otherwise there would be a loop in $G$ which cannot be contracted in any simplicial complex with $G$ as its $1$-dimensional skeleton. 
Denote by $C_1$ and $C_2$ the connected components of $e$ and $h$, respectively in $G \backslash T_i$. Since the left multiplications are graph isomorphisms, removing $ g_i^{-1}T_i$ disconnects $G$ in two components $C_1^{\prime}$ and $C_2^{\prime}$ isomorphic to $C_1$ and $C_2$ respectively. If $g_i^{-1} T_i$ does not intersect $T_i$, then one of $C_1^{\prime}$ or $C_2^{\prime}$ properly contains $C_2$, but since $\vert C_2^{\prime} \vert = \vert C_2 \vert$, we have $C_1^{\prime } \subsetneq C_2$ and $\vert C_1 \vert > \vert C_2 \vert$. The same way, if $hg^{-1}_iT_i$ does not intersect $T_i$, then $\vert C_2 \vert > \vert C_1 \vert$.
Therefore $T_i$ has to intersect either $g_i^{-1}T_i$ or $hg_i^{-1}T_i$, so $\vert T_i \vert \geq diam(T_i) \geq \min \{   i, diam(G) -i  \} +1$. Since the $T_i$'s are disjoint, summing over all $i$'s yield
$$\vert G \vert \geq \frac{(diam(G)+2)^2 +1}{4}.$$
Which is the desired inequality.