MathDBThe quasi-isometric rigidity conjecture for simply connected nilpotent Lie groups
The quasi-isometric rigidity conjecture for simply connected nilpotent Lie groups
Let and be simply connected nilpotent Lie groups. Quasi-isometric rigidity conjecture. The groups and are quasi-isometric if and only if they are isomorphic. This would give a neat formulation of the quasi-isometry classification of finitely generated nilpotent groups, which reduces to the classification of their real Malcev completions. The conjecture is cited as a central open problem in the large-scale geometry of nilpotent groups.
Equivalent formulations 1
Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.
The quasi-isometric rigidity conjecture for simply-connected nilpotent Lie groups
Let and be simply-connected nilpotent Lie groups. They are quasi-isometric if their underlying metric spaces are quasi-isometric.
Quasi-isometric rigidity conjecture. Quasi-isometric simply-connected nilpotent Lie groups are isomorphic.
This is a well-known conjecture in the quasi-isometric classification of nilpotent Lie groups. The paper provides evidence by constructing new rigid examples, but the general statement remains open.
source: Manuel Amann, “On quasi-isometric nilpotent Lie groups”, arXiv:1710.04542 (2017).
Sources & referencesView supporting material
Primary source
Claudio Llosa Isenrich, Gabriel Pallier and Romain Tessera, “Cone-equivalent nilpotent groups with different Dehn functions”, arXiv:2008.01211 (2023).
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