  We study random surfaces constructed by glueing together $N/k$ filled
$k$-gons along their edges, with all $(N-1)!! = (N-1)(N-3)...3\cdot 1$ pairings
of the edges being equally likely. (We assume that lcm $\{2,k\}$ divides $N$.)
The Euler characteristic of the resulting surface is related to the number of
cycles in a certain random permutation of $\{1, ..., N\}$. Gamburd has shown
that when 2 lcm $\{2,k\}$ divides $N$, the distribution of this random
permutation converges to that of the uniform distribution on the alternating
group $A_N$ in the total-variation distance as $N\to\infty$. We obtain
large-deviations bounds for the number of cycles that, together with Gamburd's
result, allow us to derive sharp estimates for the moments of the number of
cycles. These estimates allow us to confirm certain cases of conjectures made
by Pippenger and Schleich.
