Horizontal joinability in canonical 3-step Carnot groups with corank 2 horizontal distributions
type: Item
instance of: scholarly article
title: Horizontal joinability in canonical 3-step Carnot groups with corank 2 horizontal distributions
zbMATH Open document ID: 1484.53060
DOI: 10.1134/S0037446621040030
published in: Siberian Mathematical Journal
publication date: 2021-09-27T00:00:00Z
review text: Suppose \(G\) is a Carnot group with Lie algebra \(\bigoplus_{i=1}^r V_i\), and write \(V_1 = \text{span} \{ X_1,\dots,X_{\dim V_1} \}\) as its first (horizontal) layer. Given any vector \(X \in V_1\) with \(X \neq 0\) and any point \(p \in G\), we call a set of the form \[ \{ p * \text{exp}(sX) \, : s \in [0,1] \} \subset G \] a horizontal segment. The authors of the paper under review define a horizontal broken line to be any connected set consisting of the union of finitely many horizontal segments which pairwise intersect only at their endpoints. The famous Rashevskii-Chow Theorem guarantees that any two points in \(G\) can be connected by a horizontal broken line consisting of at most \(\dim G \cdot (3 \cdot 2^{r-1} - 2)\) horizontal segments where \(\dim G\) is the topological dimension of \(G\). In the paper under review, the authors consider the question of the minimal number \(N\) of horizontal segments required to connect two arbitrary points in a Carnot group. It was previously proven by the first author that this number is 3 in the Heisenberg group and 4 in the Engel group. The authors here prove that the number \(N\) in a 2-step Carnot group with a corank-1 horizontal distribution is 3, and the number \(N\) in any canonical 3-step Carnot group with a corank-2 horizontal distribution is at most 7.
reviewed by: Scott Zimmerman
Mathematics Subject Classification ID: 53C17
Mathematics Subject Classification ID: 22E25
zbMATH DE Number: 7401157
zbMATH Keywords: left-invariant basis vector fields
zbMATH Keywords: horizontal broken line
zbMATH Keywords: Rashevskii-Chow theorem
zbMATH Keywords: Carnot group
author: A. V. Greshnov
author: Roman Ivanovich Zhukov
MaRDI profile type: Publication
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full work available at URL: s0037446621040030
OpenAlex ID: W3197563134
description: scientific article; zbMATH DE number 7401157