Horizontal joinability in canonical 3-step Carnot groups with corank 2 horizontal distributions
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.
scientific article; zbMATH DE number 7401157
scientific article