  We study the game of Cops and Robbers, where cops try to capture a robber on
the vertices of a graph. Meyniel's conjecture states that for every connected
graph $G$ on $n$ vertices, the cop number of $G$ is upper bounded by
$O(\sqrt{n})$, i.e., that $O(\sqrt{n})$ suffice to catch the robber. We present
several families of abelian Cayley graphs that are Meyniel extremal, i.e.,
graphs whose cop number is $O(\sqrt{n})$. This proves that the $O(\sqrt{n})$
upper bound for Cayley graphs proved by Bradshaw is tight up to a
multiplicative constant. In particular, this shows that Meyniel's conjecture,
if true, is tight to a multiplicative constant even for abelian Cayley graphs.
  In order to prove the result, we construct Cayley graphs on $n$ vertices with
$\Omega(\sqrt{n})$ generators that are $K_{2,3}$-free. This shows that the
K\"{o}v\'{a}ri, S\'{o}s, and Tur\'{a}n theorem, stating that any $K_{2,3}$-free
graph of $n$ vertices has at most $O(n^{3/2})$ edges, is tight up to a
multiplicative constant even for abelian Cayley graphs.
