We consider left-invariant distances $d$ on a Lie group $G$ with the property that there exists a multiplicative one-parameter group of Lie automorphisms $(0, \infty)\rightarrow\mathtt{Aut}(G)$, $\lambda\mapsto\delta_\lambda$, so that $ d(\delta_\lambda x,\delta_\lambda y) = \lambda d(x,y)$, for all $x,y\in G$ and all $\lambda>0$. First, we show that all such distances are admissible, that is, they induce the manifold topology. Second, we characterize multiplicative one-parameter groups of Lie automorphisms that are dilations for some left-invariant distance in terms of algebraic properties of their infinitesimal generator. Third, we show that an admissible left-invariant distance on a Lie group with at least one nontrivial dilating automorphism is biLipschitz equivalent to one that admits a one-parameter group of dilating automorphisms. Moreover, the infinitesimal generator can be chosen to have spectrum in $[1,\infty)$. Fourth, we characterize the automorphisms of a Lie group that are a dilating automorphisms for some admissible distance. Finally, we characterize metric Lie groups admitting a one-parameter group of dilating automorphisms as the only locally compact, isometrically homogeneous metric spaces with metric dilations of all factors.

We study infinitesimal and asymptotic properties of geodesics (i.e., isometric images of intervals) in Carnot groups equipped with arbitrary sub-Finsler metrics. We show that tangents of geodesics are geodesics in some groups of lower nilpotency step. Namely, every blowup curve of every geodesic in every Carnot group is still a geodesic in the group modulo its last layer. With the same approach, we also show that blowdown curves of geodesics in sub-Riemannian Carnot groups are contained in subgroups of lower rank. This latter result can be extended to rough geodesics.

We show that any continuous open surjection from a complete metric space to another metric space can be restricted to a surjection for which the domain has the same density character as the target. This improves a recent result of Aron, Jaramillo and Le Donne.

The purpose of this note is to record a consequence, for general metric spaces, of a recent result of David Bate. We prove the following fact: Let $X$ be a compact metric space of topological dimension $n$. Suppose that the $n$-dimensional Hausdorff measure of $X$, $H^n(X)$, is finite. Suppose further that the lower $n$-density of the measure $H^n(X)$ is positive, $H^n(X)$-almost everywhere in $X$. Then $X$ contains an $n$-rectifiable subset of positive $H^n(X)$-measure. Moreover, the assumption on the lower density is unnecessary if one uses recently announced results of Csörnyei-Jones.

We study locally compact metric spaces that enjoy various forms of homogeneity with respect to Möbius self-homeomorphisms. We investigate connections between such homogeneity and the combination of isometric homogeneity with invertibility. In particular, we provide a new characterization of snowflakes of boundaries of rank-one symmetric spaces of non-compact type among locally compact and connected metric spaces. Furthermore, we investigate the metric implications of homogeneity with respect to uniformly strongly quasi-Möbius self-homeomorphisms, connecting such homogeneity with the combination of uniform bi-Lipschitz homogeneity and quasi-invertibility. In this context we characterize spaces containing a cut point and provide several metric properties of spaces containing no cut points. These results are motivated by a desire to characterize the snowflakes of boundaries of rank-one symmetric spaces up to bi-Lipschitz equivalence.

This note is concerned with the geometric classification of connected Lie groups of dimension three or less, endowed with left-invariant Riemannian metrics. On the one hand, assembling results from the literature, we give a review of the complete classification of such groups up to quasi-isometries and we compare the quasi-isometric classification with the bi-Lipschitz classification. On the other hand, we study the problem whether two quasi-isometrically equivalent Lie groups may be made isometric if equipped with suitable left-invariant Riemannian metrics. We show that this is the case for three-dimensional simply connected groups, but it is not true in general for multiply connected groups. The counterexample also demonstrates that `may be made isometric' is not a transitive relation.

This paper contributes to the generalization of Rademacher's differentiability result for Lipschitz functions when the domain is infinite dimensional and has nonabelian group structure. We introduce the notion of metric scalable groups which are our infinite-dimensional analogues of Carnot groups. The groups in which we will mostly be interested are the ones that admit a dense increasing sequence of (finite-dimensional) Carnot subgroups. In fact, in each of these spaces we show that every Lipschitz function has a point of G\^{a}teaux differentiability. We provide examples and criteria for when such Carnot subgroups exist. The proof of the main theorem follows the work of Aronszajn and Pansu.

The image of the branch set of a PL branched cover between PL $n$-manifolds is a simplicial $(n-2)$-complex. We demonstrate that the reverse implication also holds; i.e., for a branched cover $f \colon \mathbb{S}^n \to \mathbb{S}^n$ with the image of the branch set contained in a simplicial $(n-2)$-complex the mapping can be reparametrized as a PL mapping. This extends a result by Martio and Srebro [MS79].

For all $n \geq 2$, we construct a metric space $(X,d)$ and a quasisymmetric mapping $f\colon [0,1]^n \rightarrow X$ with the property that $f^{-1}$ is not absolutely continuous with respect to the Hausdorff $n$-measure on $X$. That is, there exists a Borel set $E \subset [0,1]^n$ with Lebesgue measure $|E|>0$ such that $f(E)$ has Hausdorff $n$-measure zero. The construction may be carried out so that $X$ has finite Hausdorff $n$-measure and $|E|$ is arbitrarily close to 1, or so that $|E|=1$. This gives a negative answer to a question of Heinonen and Semmes.

We construct quasiconformal mappings $f\colon \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ for which there is a Borel set $E \subset \mathbb{R}^2 \times \{0\}$ of positive Lebesgue $2$-measure whose image $f(E)$ has Hausdorff $2$-measure zero. This gives a solution to the open problem of inverse absolute continuity of quasiconformal mappings on hypersurfaces, attributed to Gehring. By implication, our result also answers questions of Väisälä and Astala--Bonk--Heinonen.

We prove that any metric space $X$ homeomorphic to $\mathbb{R}^2$ with locally finite Hausdorff 2-measure satisfies a reciprocal lower bound on modulus of curve families associated to a quadrilateral. More precisely, let $Q \subset X$ be a topological quadrilateral with boundary edges (in cyclic order) denoted by $ζ_1, ζ_2, ζ_3, ζ_4$ and let $Γ(ζ_i, ζ_j; Q)$ denote the family of curves in $Q$ connecting $ζ_i$ and $ζ_j$; then $\text{mod } Γ(ζ_1, ζ_3; Q) \text{mod } Γ(ζ_2, ζ_4; Q) \geq 1/κ$ for $κ= 2000^2\cdot (4/π)^2$. This answers a question concerning minimal hypotheses under which a metric space admits a quasiconformal parametrization by a domain in $\mathbb{R}^2$.

We define a Newman property for BLD-mappings and study its connections to the porosity of the branch set in the setting of generalized manifolds equipped with complete path metrics.

Sto\"ilow's theorem from 1928 states that a continuous, light, and open mapping between surfaces is a discrete map with a discrete branch set. This result implies that such mappings between orientable surfaces are locally modelled by power mappings $z\mapsto z^k$ and admit a holomorphic factorization. The purpose of this expository article is to give a proof of this classical theorem having the readers interested in discrete and open mappings in mind.