We characterize Carnot groups admitting a 1-quasiconformal metric inversion as the Lie groups of Heisenberg type whose Lie algebras satisfy the J2-condition, thus characterizing a special case of inversion invariant bi-Lipschitz homogeneity. A more general characterization of inversion invariant bi-Lipschitz homogeneity for certain non-fractal metric spaces is also provided.

In this article we study the Ahlfors regular conformal gauge of a compact metric space $(X,d)$, and its conformal dimension ${dim}_{AR}(X,d)$. Using a sequence of finite coverings of $(X,d)$, we construct distances in its Ahlfors regular conformal gauge of controlled Hausdorff dimension. We obtain in this way a combinatorial description, up to bi-Lipschitz homeomorphisms, of all the metrics in the gauge. We show how to compute ${dim}_{AR}(X,d)$ using the critical exponent $Q_N$ associated to the combinatorial modulus.

By $F_n\left(X\right)$, $n\ge 1$, we denote the $n$-th symmetric product of a metric space $(X,d)$ as the space of the non-empty finite subsets of $X$ with at most $n$ elements endowed with the Hausdorff metric $d_H$. In this paper we shall describe that every isometry from the $n$-th symmetric product $F_n\left(X\right)$ into itself is induced by some isometry from $X$ into itself, where $X$ is either the Euclidean space or the sphere with the usual metrics. Moreover, we study the $n$-th symmetric product of the Euclidean space up to bi-Lipschitz equivalence and...

We consider sets in uniformly perfect metric spaces which are null for every doubling measure of the space or which have positive measure for all doubling measures. These sets are called thin and fat, respectively. In our main results, we give sufficient conditions for certain cut-out sets being thin or fat.

We show that the tangent cone at the identity is not a complete quasiconformal invariant for sub-Riemannian nilpotent groups. Namely, we show that there exists a nilpotent Lie group equipped with left invariant sub-Riemannian metric that is not locally quasiconformally equivalent to its tangent cone at the identity. In particular, such spaces are not locally bi-Lipschitz homeomorphic. The result is based on the study of Carnot groups that are rigid in the sense that their only quasiconformal maps...

Let (Z, d, μ) be a compact, connected, Ahlfors Q-regular metric space with Q > 1. Using a hyperbolic filling of Z,we define the notions of the p-capacity between certain subsets of Z and of theweak covering p-capacity of path families Γ in Z.We show comparability results and quasisymmetric invariance.As an application of our methodswe deduce a result due to Tyson on the geometric quasiconformality of quasisymmetric maps between compact, connected Ahlfors Q-regular metric spaces.