  Random hyperbolic graphs have been suggested as a promising model of social
networks. A few of their fundamental parameters have been studied. However,
none of them concerns their spectra. We consider the random hyperbolic graph
model as formalized by [GPP12] and essentially determine the spectral gap of
their normalized Laplacian. Specifically, we establish that with high
probability the second smallest eigenvalue of the normalized Laplacian of the
giant component of and $n$-vertex random hyperbolic graph is
$\Omega(n^{-(2\alpha-1)}/D)$, where $\frac12<\alpha<1$ is a model parameter and
$D$ is the network diameter (which is known to be at most polylogarithmic in
$n$). We also show a matching (up to a polylogarithmic factor) upper bound of
$n^{-(2\alpha-1)}(\log n)^{1+o(1)}$. As a byproduct we conclude that the
conductance upper bound on the eigenvalue gap obtained via Cheeger's inequality
is essentially tight. We also provide a more detailed picture of the collection
of vertices on which the bound on the conductance is attained, in particular
showing that for all subsets whose volume is $O(n^{1-\varepsilon})$ the
obtained conductance is with high probability
$\Omega(n^{-(2\alpha-1)\varepsilon+o(1)})$. Finally, we also show consequences
of our result for the minimum and maximum bisection of the giant component.

Probability;Combinatorics