Related papers: Sharp measure contraction property for generalized…

We prove that any corank 1 Carnot group of dimension $k+1$ equipped with a left-invariant measure satisfies the $\mathrm{MCP}(K,N)$ if and only if $K \leq 0$ and $N \geq k+3$. This generalizes the well known result by Juillet for the…

We initiate the study of synthetic curvature-dimension bounds in sub-Finsler geometry. More specifically, we investigate the measure contraction property $\mathsf{MCP}(K, N)$, and the geodesic dimension on the Heisenberg group equipped with…

Rank-one symmetric spaces carry a solvable group model which have a generalization to a larger class of Lie groups that are one-dimensional extensions of nilpotent groups. By examining some metric properties of these symmetric spaces, we…

In Carnot groups of step 2 we consider sets having maximal or minimal possible homogeneous Hausdorff dimension compared to their Euclidean one: in the first case we prove that they must be in a sense vertical, that is a large part of these…

A group $H \cong {\mathbb Z}_{2}^{n}$, $n \geq 3$, of conformal automorphisms of a closed Riemann surface $S$ such that $S/H$ has genus zero and exactly $(n+1)$ cone points is called a generalized Humbert group of type $n$, in which case,…

We show that every graded nilpotent Lie group $G$ of step $r$, equipped with a left invariant metric homogeneous with respect to the dilations induced by the grading, (this includes all Carnot groups with Carnot-Caratheodory metric) is…

This paper explicitly constructs the complete set of optimal sub-Riemannian geodesics starting from a point for certain Carnot groups of step two. These are groups of dimension 2n+1 equipped with a left-invariant distribution of dimension…

This paper is devoted to show that the flatness of tangents of $1$-codimensional measures in Carnot Groups implies $C^1_\mathbb{G}$-rectifiability. As applications we prove that measures with $(2n+1)$-density in the Heisenberg groups…

The surface Houghton groups $\mathcal{H}_{n}$ are a family of groups generalizing Houghton groups $H_n$, which are constructed as asymptotically rigid mapping class groups. We give a complete computation of the BNSR-invariants…

In this paper, we construct H\"older maps to Carnot groups equipped with a Carnot metric, especially the first Heisenberg group $\mathbb{H}$. Pansu and Gromov observed that any surface embedded in $\mathbb{H}$ has Hausdorff dimension at…

Measure contraction properties $MCP(K,N)$ are synthetic Ricci curvature lower bounds for metric measure spaces which do not necessarily have smooth structures. It is known that if a Riemannian manifold has dimension $N$, then $MCP(K,N)$ is…

We provide a new and elementary proof for the structure of geodesics in the Heisenberg group $\mathbb{H}^n$. The proof is based on a new isoperimetric inequality for closed curves in $\mathbb{R}^{2n}$. We also prove that the…

Let $\mathcal{S}_H^0(K)$, $K\ge 1$, be the class of normalized $K$-quasiconformal harmonic mappings in the unit disk. We obtain Baernstein type extremal results for the analytic and co-analytic parts of functions in the geometric subclasses…

This paper aims to study isometries of the $1$-Wasserstein space $\mathcal{W}_1(\mathbf{G})$ over Carnot groups endowed with horizontally strictly convex norms. Well-known examples of horizontally strictly convex norms on Carnot groups are…

In this article, we prove that the neighborhood complex of the Kneser graph $KG_{3,k}$ is of the same homotopy type as that of a wedge of $\frac{(k+1)(k+3)(k+4)(k+6)}{4}+1$ spheres of dimension $k$. We construct a maximal subgraph $S_{3,k}$…

In this paper we characterize the geodesic dimension $N_{GEO}$ and give a new lower bound to the curvature exponent $N_{CE}$ on Sard-regular Carnot groups. As an application, we give an example of step-two Carnot group on which $N_{CE} >…

The H-type deviation, $\delta({\mathbb G})$, of a step two Carnot group ${\mathbb G}$ quantifies the extent to which ${\mathbb G}$ deviates from the geometrically and algebraically tractable class of Heisenberg-type (H-type) groups. In an…

In this note we show that in metric measure spaces satisfying the reduced curvature-dimension condition CD*(K,N) we always have geodesics in the Wasserstein space of probability measures that satisfy the critical convexity inequality of…

For any $n\geq k\geq l\in\mathbb{N},$ let $S(n,k,l)$ be the set of all those non-negative definite matrices $a\in M_{n}(\mathbb{C})$ with $l\leq\text{rank }a\leq k$. Motivated by applications to $C^{*}$-algebra theory, we investigate the…

In the study of the generalization of Hilbert's Third Problem to spherical geometry, Sah constructed a Hopf algebra of spherical polytopes with product given by join and coproduct given by a generalized Dehn invariant. Using Zakharevich's…