  We study properties of typical closed geodesics on expander surfaces of high
genus, i.e. closed hyperbolic surfaces with a uniform spectral gap of the
Laplacian. Under an additional systole lower bound assumption, we show almost
every geodesic of length much greater than $\sqrt{g}\log g$ is non-simple. And
we prove almost every closed geodesic of length much greater than $g (\log
g)^2$ is filling, i.e. each component of the complement of the geodesic is a
topological disc. Our results apply to Weil-Petersson random surfaces, random
covers of a fixed surface, and Brooks-Makover random surfaces, since these
models are known to have uniform spectral gap asymptotically almost surely.
  Our proof technique involves adapting Margulis' counting strategy to work at
low length scales.
