Quasicircle Boundaries and Exotic Almost-isometries

We consider properly discontinuous, isometric, convex cocompact actions of surface groups Γ on a CAT(-1) space X. 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 biLipschitz 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 H 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 familes of equal area Riemannian metrics, with the property that their lifts to the universal covers are pairwise almostisometric but are not isometric to each other. Finally, we exhibit a gap phenomena for the optimal multiplicative constant for a quasi-isometry between periodic CAT(-1) metrics on H.