Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access March 30, 2021

Sub-Finsler Horofunction Boundaries of the Heisenberg Group

  • Nate Fisher and Sebastiano Nicolussi Golo EMAIL logo


We give a complete analytic and geometric description of the horofunction boundary for polygonal sub-Finsler metrics, that is, those that arise as asymptotic cones of word metrics, on the Heisenberg group. We develop theory for the more general case of horofunction boundaries in homogeneous groups by connecting horofunctions to Pansu derivatives of the distance function.

MSC 2010: 20F69; 53C23; 53C17


[1] Uri Bader and Vladimir Finkelshtein. On the horofunction boundary of discrete Heisenberg group. Geometriae Dedicata, pages 1–15, 2020.10.1007/s10711-020-00513-xSearch in Google Scholar

[2] Martin R. Bridson and André Haefliger. Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999.10.1007/978-3-662-12494-9Search in Google Scholar

[3] Herbert Busemann. The isoperimetric problem in the Minkowski plane. Amer. J. Math., 69:863–871, 1947.10.2307/2371807Search in Google Scholar

[4] Luca Capogna, Donatella Danielli, Scott D. Pauls, and Jeremy T. Tyson. An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem, volume 259 of Progress in Mathematics. Birkhäuser Verlag, Basel, 2007.Search in Google Scholar

[5] Corina Ciobotaru, Linus Kramer, and Petra Schwer. Polyhedral compactifications, i. arXiv preprint arXiv:2002.12422, 2020.Search in Google Scholar

[6] Moon Duchin and Nathan Fisher. Stars at infinity in Teichmüller space. arXiv preprint arXiv:2004.04231, 2020.Search in Google Scholar

[7] Moon Duchin and Christopher Mooney. Fine asymptotic geometry in the Heisenberg group. Indiana Univ. Math. J., 63(3):885–916, 2014.10.1512/iumj.2014.63.5308Search in Google Scholar

[8] G. B. Folland and Elias M. Stein. Hardy spaces on homogeneous groups, volume 28 of Mathematical Notes. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982.10.1515/9780691222455Search in Google Scholar

[9] V. Gershkovich and A. Vershik. Nonholonomic manifolds and nilpotent analysis. J. Geom. Phys., 5(3):407–452, 1988.10.1016/0393-0440(88)90032-0Search in Google Scholar

[10] Lizhen Ji and Anna-Sofie Schilling. Polyhedral horofunction compactification as polyhedral ball. arXiv preprint arXiv:1607.00564, 2016.Search in Google Scholar

[11] Antoine Julia, Sebastiano Nicolussi Golo, and Davide Vittone. Area of intrinsic graphs and coarea formula in Carnot Groups. arXiv e-prints, page arXiv:2004.02520, April 2020.Search in Google Scholar

[12] Anders Karlsson, Volker Metz, and Gennady A Noskov. Horoballs in simplices and Minkowski spaces. International journal of mathematics and mathematical sciences, 2006, 2006.10.1155/IJMMS/2006/23656Search in Google Scholar

[13] Tom Klein and Andrew Nicas. The horofunction boundary of the Heisenberg group. Pacific journal of mathematics, 242(2):299–310, 2009.10.2140/pjm.2009.242.299Search in Google Scholar

[14] Tom Klein and Andrew Nicas. The horofunction boundary of the Heisenberg group: the Carnot-Carathéodory metric. Conformal Geometry and Dynamics of the American Mathematical Society, 14(15):269–295, 2010.Search in Google Scholar

[15] Enrico Le Donne. A primer on Carnot groups: homogenous groups, Carnot-Carathéodory spaces, and regularity of their isometries. Anal. Geom. Metr. Spaces, 5(1):116–137, 2017.Search in Google Scholar

[16] Enrico Le Donne and Sebastiano Nicolussi Golo. Regularity properties of spheres in homogeneous groups. Trans. Amer. Math. Soc., 370(3):2057–2084, 2018.10.1090/tran/7038Search in Google Scholar

[17] Enrico Le Donne, Sebastiano Nicolussi Golo, and Andrea Sambusetti. Asymptotic behavior of the Riemannian Heisenberg group and its horoboundary. Ann. Mat. Pura Appl. (4), 196(4):1251–1272, 2017.10.1007/s10231-016-0615-2Search in Google Scholar

[18] Enrico Le Donne and Séverine Rigot. Besicovitch covering property on graded groups and applications to measure differentiation. J. Reine Angew. Math., 750:241–297, 2019.10.1515/crelle-2016-0051Search in Google Scholar

[19] John Mitchell. On Carnot-Carathéodory metrics. J. Differential Geom., 21(1):35–45, 1985.10.4310/jdg/1214439462Search in Google Scholar

[20] John William Mitchell. A local study of Carnot-Caratheodory metrics. ProQuest LLC, Ann Arbor, MI, 1982. Thesis (Ph.D.)–State University of New York at Stony Brook.Search in Google Scholar

[21] Alexander Nagel, Elias M. Stein, and Stephen Wainger. Balls and metrics defined by vector fields. I. Basic properties. Acta Math., 155(1-2):103–147, 1985.10.1007/BF02392539Search in Google Scholar

[22] Pierre Pansu. Croissance des boules et des géodésiques fermées dans les nilvariétés. Ergodic Theory Dynam. Systems, 3(3):415–445, 1983.10.1017/S0143385700002054Search in Google Scholar

[23] Pierre Pansu. Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. Ann. of Math. (2), 129(1):1–60, 1989.10.2307/1971484Search in Google Scholar

[24] Cormac Walsh. The horofunction boundary of finite-dimensional normed spaces. Math. Proc. Cambridge Philos. Soc., 142(3):497–507, 2007.10.1017/S0305004107000096Search in Google Scholar

[25] Cormac Walsh. The action of a nilpotent group on its horofunction boundary has finite orbits. Groups Geom. Dyn., 5(1):189–206, 2011.10.4171/GGD/122Search in Google Scholar

[26] Kazimierz Zarankiewicz. Sur les points de division dans les ensembles connexes. Uniwersytet, Seminarjum Matematyczne, 1927.10.4064/fm-9-1-124-171Search in Google Scholar

Received: 2020-10-24
Accepted: 2021-01-13
Published Online: 2021-03-30

© 2021 Nate Fisher et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Downloaded on 28.2.2024 from
Scroll to top button