  We consider properly discontinuous, isometric, convex cocompact actions of
surface groups on a CAT(-1) space. We show that the limit set of such an
action, equipped with the canonical visual metric, is a (weak) quasicircle in
the sense of Falconer and Marsh. It follows that the visual metrics on such
limit sets are classified, up to bi-Lipschitz equivalence, by their Hausdorff
dimension. This result applies in particular to boundaries at infinity of the
universal cover of a locally CAT(-1) surface. We show that any two periodic
CAT(-1) metrics on $\mathbb H^2$ can be scaled so as to be almost-isometric
(though in general, no equivariant almost-isometry exists). We also construct,
on each higher genus surface, $k$-dimensional families of equal area Riemannian
metrics, with the property that their lifts to the universal covers are
pairwise almost-isometric but are not isometric to each other. Finally, we
exhibit a gap phenomenon for the optimal multiplicative constant for a
quasi-isometry between periodic CAT(-1) metrics on $\mathbb H^2$.
