In [4], M. Bonk and B. Kleiner proved a rigidity theorem for expanding quasi-Möbius group actions on Ahlfors n-regular metric spaces with topological dimension n. This led naturally to a rigidity result for quasi-convex geometric actions on CAT(−1)-spaces that can be seen as a metric analog to the "entropy rigidity" theorems of U. Hamenstädt [16] and M. Bourdon [8]. Building on the ideas developed in [4], we establish a rigidity theorem for certain expanding quasi-Möbius group actions on spaces with different metric and topological dimensions. This is motivated by a corresponding entropy rigidity result in the coarse geometric setting.

Abstract. A theorem of W. Derrick ensures that the volume of any Riemannian cube ([0, 1] n , g) is bounded below by the product of the distances between opposite codimension-1 faces. In this paper, we establish a discrete analog of Derrick's inequality for weighted open covers of the cube [0, 1] n , which is motivated by a question about lower volume bounds in metric spaces. Our main theorem generalizes a previous result of the author in [13], which gave a combinatorial version of Derrick's inequality and was used in the analysis of boundaries of hyperbolic groups. As an application, we answer a question of Y. Burago and V. Zalgaller about length-volume inequalities for pseudometrics on the unit cube.

Building off of techniques that were recently developed by M. Carrasco, S. Keith, and B. Kleiner to study the conformal dimension of boundaries of hyperbolic groups, we prove that uniformly perfect boundaries of John domains inĈ have conformal dimension equal to 0 or 1. Our proof uses a discretized version of Carrasco's "uniformly well-spread cut point" condition, which we call the discrete UWS property, that is well-suited to deal with metric spaces that are not linearly connected. More specifically, we prove that boundaries of John domains have the discrete UWS property and that any compact, doubling, uniformly perfect metric space with the discrete UWS property has conformal dimension equal to 0 or 1. In addition, we establish other geometric properties of metric spaces with the discrete UWS property, including connectivity properties of their weak tangents.Theorem 1.1. If Ω ⊂Ĉ is a John domain with ∂Ω uniformly perfect, then Cdim AR (∂Ω) ∈ {0, 1}. It is is equal to 0 if and only if ∂Ω is uniformly disconnected.We should note that, in general, Cdim(X) ≤ Cdim AR (X). Thus, we also obtain Cdim(∂Ω) ∈ {0, 1}, as the conformal dimension cannot assume values in (0, 1) by the results in [17]. If we assume in addition that ∂Ω is connected and not a singleton, then this gives Cdim(∂Ω) = Cdim AR (∂Ω) = 1.Before moving on, let us mention a corollary that deals with the quasiconformal geometry of certain Julia sets. We noted above that if p(z) is a semi-hyperbolic polynomial onĈ of degree at least 2, then its unbounded Fatou component is a John domain [7]. The Julia set J(p) of p is the boundary of this component, so Theorem 1.1 implies the following.Corollary 1.2. If p(z) is a semi-hyperbolic polynomial of degree at least 2 with J(p) connected, then Cdim(J(p)) = Cdim AR (J(p)) = 1.

We establish Euclidean-type lower bounds for the codimension-1 Hausdorff measure of sets that separate points in doubling and linearly locally contractible metric manifolds. This gives a quantitative topological isoperimetric inequality in the setting of metric manifolds, in the sense that lower bounds for the codimension-1 measure of a set depend not on some notion of filling or volume but rather on in-radii of complementary components. As a consequence, we show that balls in a closed, connected, doubling, and linearly locally contractible metric n-manifold (M, d) with radius 0 < r ≤ diam(M ) have n-dimensional Hausdorff measure at least c · r n , where c > 0 depends only on n and on the doubling and linear local contractibility constants.

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