Geodesic
Box-quasimetrics and horizontal joinability on Cartan groups
Algebra i logika, Tome 63 (2024) no. 1, pp. 15-29

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On a Cartan group $\mathbb K$ equipped with a Carnot–Carathéodory metric $d_{cc}$, we find the exact value of a constant in the $(1,q_2)$-generalized triangle inequality for its Box-quasimetric. It is proved that any two points $x,y\in\Bbb K$ can be joined by a horizontal $k$-broken line $L^k_{x,y}$, $k\leq 6$; moreover, the length of such a broken line $L^k_{x,y}$ does not exceed the quantity $Cd_{cc}(x,y)$ for some constant $C$ not depending on the choice of $x,y\in\mathbb K$. The value $6$ here is nearly optimal.
Mots-clés : Cartan group, $(q_1,q_2)$-quasimetric space, Box-quasimetric
Keywords: horizontal broken line, Rashevskii–Chow theorem. 
@article{AL_2024_63_1_a1,
     author = {A. V. Greshnov and V. S. Kostyrkin},
     title = {Box-quasimetrics and horizontal joinability on {Cartan} groups},
     journal = {Algebra i logika},
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     publisher = {mathdoc},
     volume = {63},
     number = {1},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2024_63_1_a1/}
}
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A. V. Greshnov; V. S. Kostyrkin. Box-quasimetrics and horizontal joinability on Cartan groups. Algebra i logika, Tome 63 (2024) no. 1, pp. 15-29. http://geodesic.mathdoc.fr/item/AL_2024_63_1_a1/