Ballmann's higher rank rigidity conjecture for CAT(0) spaces

Let XX be a locally compact CAT(0) space with a geometric group action ΓX\Gamma\curvearrowright X. An nn-flat is an isometrically embedded copy of Euclidean nn-space, where n2n\geq 2. Higher rank rigidity conjecture. If every geodesic in XX lies in an nn-flat, then XX is a Riemannian symmetric space, a Euclidean building, or non-trivially splits as a metric product. This conjecture is the CAT(0)-space analogue of higher-rank rigidity for Hadamard manifolds; the supplied source presents it as the main motivation and does not state whether it is resolved in this generality.

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Primary source

Stephan Stadler, “CAT(0) spaces of higher rank II”, arXiv:2212.07092 (2022).

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