## On the logarithmic coarse structures of Lie groups and hyperbolic spaces

created by pallier1 on 24 May 2021
modified on 02 Aug 2022

[BibTeX]

preprint

Inserted: 24 may 2021
Last Updated: 2 aug 2022

Journal: Geometriae Dedicata
Volume: 216
Number: 56
Year: 2022

ArXiv: 2105.03955 PDF
We characterize the Lie groups with finitely many connected components that are $O(u)$-bilipschitz equivalent (almost quasiisometric in the sense that the sublinear function $u$ replaces the additive bounds of quasiisometry) to the real hyperbolic space, or to the complex hyperbolic plane. The characterizations are expressed in terms of deformations of Lie algebras and in terms of pinching of sectional curvature of left-invariant Riemannian metrics in the real case. We also compare sublinear bilipschitz equivalence and coarse equivalence, and prove that every coarse equivalence between the logarithmic coarse structures of geodesic spaces is a $O(\log)$-bilipschitz equivalence. The Lie groups characterized are exactly those whose logarithmic coarse structure is equivalent to that of a real hyperbolic space or the complex hyperbolic plane.