Direct limits of infinite-dimensional Carnot groups


  • Terhi Moisala
  • Enrico Pasqualetto



We give a construction of direct limits in the category of complete metric scalable groups and provide sufficient conditions for the limit to be an infinite-dimensional Carnot group. We also prove a Rademacher-type theorem for such limits.


Aronszajn, N., Differentiability of Lipschitzian mappings between Banach spaces, Studia Math. 57 (1976), no. 2, 147–190.

Baudoin, F., Gordina, M., and Melcher, T., Quasi-invariance for heat kernel measures on sub-Riemannian infinite-dimensional Heisenberg groups, Trans. Amer. Math. Soc. 365 (2013), no. 8, 4313–4350.

Driver, B. K., and Gordina, M., Heat kernel analysis on infinite-dimensional Heisenberg groups, J. Funct. Anal. 255 (2008), no. 9, 2395–2461.

Glöckner, H., Direct limits of infinite-dimensional Lie groups compared to direct limits in related categories, J. Funct. Anal. 245 (2007), no. 1, 19–61.

Grong, E., Markina, I., and Vasil'ev, A., Sub-Riemannian geometry on infinite-dimensional manifolds, J. Geom. Anal. 25 (2015), no. 4, 2474–2515.

Grong, E., Nilssen, T., and Schmeding, A., Geometric rough paths on infinite dimensional spaces, arXiv:2006.06362

Lang, S., Algebra, second edition, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1984.

Le Donne, E., A metric characterization of Carnot groups, Proc. Amer. Math. Soc. 143 (2015), no. 2, 845–849.

Le Donne, E., A primer on Carnot groups: homogeneous groups, Carnot-Carathéodory spaces, and regularity of their isometries, Anal. Geom. Metr. Spaces 5 (2017), no. 1, 116–137.

Le Donne, E., Li, S., and Moisala, T., Infinite-Dimensional Carnot Groups and Gâteaux Differentiability, J. Geom. Anal. 31 (2021), no. 2, 1756–1785.

Le Donne, E., and Züst, R., Space of signatures as inverse limits of Carnot groups, ESAIM Control Optim. Calc. Var. 27 (2021), Paper No. 37, 14 pp.

Melcher, T., Heat kernel analysis on semi-infinite Lie groups, J. Funct. Anal. 257 (2009), no. 11, 3552–3592.

Mac Lane, S., Categories for the working mathematician, second edition, Graduate Texts in Mathematics Vol. 5, Springer-Verlag, New York, 1998.

Magnani, V., Pinamonti, A., and Speight, G., Porosity and differentiability of Lipschitz maps from stratified groups to Banach homogeneous groups, Ann. Mat. Pura Appl. (4) 199 (2020), no. 3, 1197–1220.

Magnani, V., and Tapio Rajala, R., Radon–Nikodym property and area formula for Banach homogeneous group targets, Int. Math. Res. Not. IMRN 2014, no. 23, 6399–6430.

Moisala, T., Unraveling intrinsic geometry of sets and functions in Carnot groups. PhD thesis, University of Jyväskylä, 2020.

Montgomery, D., Continuity in topological groups, Bull. Amer. Math. Soc. 42 (1936), no. 12, 879–882.

Montgomery, D., and Zippin, L., Topological transformation groups, Interscience Publishers, New York-London, 1955.

Pansu, P., Métriques de Carnot-Carathéodory et quasiisométries des espaces symé­triques de rang un, Ann. of Math. (2) 129 (1989), no. 1, 1–60.

Rosendal, C., Automatic continuity of group homomorphisms, Bull. Symbolic Logic 15 (2009), no. 2, 184–214.

Tkachenko, M., Paratopological and semitopological groups versus topological groups, Recent progress in general topology. III, 825–882, Atlantis Press, Paris, 2014.



How to Cite

Moisala, T., & Pasqualetto, E. (2022). Direct limits of infinite-dimensional Carnot groups. MATHEMATICA SCANDINAVICA, 128(2).