Stratified \(\beta \)-numbers and traveling salesman in Carnot groups. (English) Zbl 1538.28012
Summary: We introduce a modified version of Jones’s \(\beta \)-numbers for Carnot groups which we call stratified \(\beta\)-numbers. We show that an analogue of Jones’s traveling salesman theorem on 1-rectifiability of sets holds for any Carnot group if we replace previous notions of \(\beta \)-numbers in Carnot groups with stratified \(\beta \)-numbers. As we generalize both directions of the traveling salesman theorem, we get a characterization of subsets of Carnot groups that lie on finite length rectifiable curves. Our proof expands upon previous analysis of the Hebisch-Sikora norm for Carnot groups. In particular, we find new estimates on the drift between almost parallel line segments that take advantage of the stratified functions \(\beta\) and also develop a Taylor expansion technique of the norm. We also give an example of a Carnot group for which a traveling salesman theorem based on the unmodified \(\beta \)-numbers must exhibit a gap between the necessary and sufficient directions.
MSC:
| 28A75 | Length, area, volume, other geometric measure theory |
| 53C17 | Sub-Riemannian geometry |
| 22E25 | Nilpotent and solvable Lie groups |
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