Quasicircle boundaries and exotic almost-isometries

TL;DR

This paper studies the geometric structure of limit sets of surface groups acting on CAT(-1) spaces, classifies visual metrics by Hausdorff dimension, and explores exotic almost-isometries between different metrics on surfaces.

Contribution

It establishes that limit sets are weak quasicircles with classification by Hausdorff dimension and constructs new examples of metrics that are almost-isometric but not isometric.

Findings

Limit sets are weak quasicircles in the sense of Falconer and Marsh.

Visual metrics are classified up to bi-Lipschitz equivalence by Hausdorff dimension.

Existence of metrics on surfaces that are pairwise almost-isometric but not isometric.

Abstract

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 $\mathbb H^2$.