Superrigidity in infinite dimension and finite rank via harmonic maps

Abstract

We prove geometric superrigidity for actions of cocompact lattices in semisimple Lie groups of higher rank on infinite dimensional Riemannian manifolds of nonpositive curvature and finite telescopic dimension.

Representations of infinite dimension orthogonal groups of quadratic forms with finite index

“…A part of this work was done during the semester "Geometric and Analytic Group Theory" at the Institute of Mathematics of the Polish Academy of Sciences and thus was partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund. rank semisimple Lie groups, a theorem similar to the geometric interpretation of Margulis superrigidity has been proved in [Duc15b,Theorem 1.2]. This gives hints that there should be no exotic such representations for higher rank semisimple Lie groups.…”

“…We claim that there is a Γ-equivariant harmonic map X G → X in the sense of Korevaar and Schoen for metric spaces. The existence of such harmonic map is provided by [Duc15b,Theorem 3.1] if we know that Γ has no fixed point in ∂X .…”

Groups acting on spaces of non-positive curvature

“…This superrigidity result inspired a lot of different works in many different directions. Let us just mention [73] for lattices in products, [40] for symmetric spaces of infinite dimension and finite rank and [48] for Busemann non-positively curved targets.…”

Hyperbolic spaces, principal series, and $${\mathrm{O}}(2,\infty )$$

“…It is interesting to study which locally compact groups admit irreducible representations into this group 1 . This question was suggested by Gromov [8, §6] and studied by Duchesne who constructed tools to establish superigidity results with target O(p, ∞) [5,6,7]. See also [3] for more recent results in this direction.…”