Abstract
A group presentation is said to have rational growth if its growth series represents a rational function, which reflects a kind of recursion in the structure of the group. A long-standing open question asks whether the Heisenberg group has rational growth for all finite generating sets, and we settle this question affirmatively. We also establish almost-convexity for all finite generating sets. Previously, both of these properties were known to hold for hyperbolic groups and virtually abelian groups, and there were no further examples in either case. Our main method is a close description of the relationship between word metrics and associated Carnot–Carathéodory Finsler metrics on the ambient Lie group. We provide (non-regular) languages of geodesics in any word metric that represent each group element uniquely.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
