Superrigidity for irreducible lattices and geometric splitting

By Nicolas Monod. 2005

123 citations recorded by the source corpus.

We prove general superrigidity results for actions of irreducible lattices on CAT(0) spaces; first, in terms of the ideal boundary, and then for the intrinsic geometry (including for infinite-dimensional spaces). In particular, one obtains a new and self-contained proof of Margulis' superrigidity theorem for uniform irreducible lattices in non-simple groups. The proofs rely on simple geometric arguments, including a splitting theorem which can be viewed as an infinite-dimensional (and singular) generalization of the Lawson–Yau/Gromoll–Wolf theorem. Appendix A gives a very elementary proof of commensurator superrigidity; Appendix B proves that all our results also hold for certain non-uniform lattices.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/70878e0bdb1767784305cd6cc9dc2150fcba8122

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Citations in the corpus

The median class and superrigidity of actions on CAT(0) cube complexes (2012, methodology): “For example, if Γ is not uniform, then in order to apply Monod’s version of Theorem 1.5, Γ has to be square-integrable and weakly cocompact.”
Isometry groups of non‐positively curved spaces: discrete subgroups (2009, methodology): “For uniform lattices, or more generally weakly cocompact squaresummable lattices, this is indeed true and was proved in [65].”
On parabolic subgroups of Artin groups (2022, unknown): “Monod proved in [Mon06, Thm. 14] that a family consisting of bounded closed convex subsets of a complete CAT(0) space with the ﬁnite intersection property has a non-empty intersection.”
Divergence and quasimorphisms of right-angled Artin groups (2010, result): “, superrigidity follows from the Margulis Normal Subgroup Theorem [Mar, Zim], since every solvable subgroup of AΓ is virtually abelian. Other special cases follow from the work of Shalom [Sha], Monod [Mon2], and Gelander–Karlsson–Margulis [GKM]. From Theorem 5.2 we also deduce the following, which P”
On the joint spectral radius for isometries of non-positively curved spaces and uniform growth (2018, unknown): “fore we start the proof, we quote some results. Let Xbe a complete CAT(0) space and @Xits visual boundary at innity. Set X= X[@X. For a closed convex subset CˆX, let CˆXbe its closure. Theorem 3.15 ([61]*Prop. 23). Assume that a complete CAT(0) space X is Gromov hyperbolic. Then for any nested famil”
Groups acting on manifolds: around the Zimmer program (2008, background): “Zimmer’s program is just one aspect of this theory, in other directions, many authors have studied homomorphisms to isometry groups of non-positively curved (and more general) metric spaces. See e.g. [23, 97, 169]. 3.3. Harmonic map approaches to rigidity. In the 90’s, ﬁrst Corlette and then Gromov-”
On the superrigidity of malleable actions with spectral gap (2006, background): “ phenomena involving product groups that have been discovered in recent years in group theory, OE ergodic theory, Borel equivalence relations and von Neumann algebras/II1 factors ([MoSh], [HK], [OP], [Mo], [BSh], etc). It would of course be interesting to ﬁnd some common ground (explanation) to thes”
At infinity of finite-dimensional CAT(0) spaces (2008, background): “Y (Xsuch that for each g∈ G, the sets Y and g.Y are at bounded Hausdorﬀ distance from one another. The relevance of the notions of evanescent and reduced actions was ﬁrst highlighted by Nicolas Monod [Mon06] in the context of geometric superrigidity. In particular, the combination of [Mon06, Theorem”
Simplicity and superrigidity of twin building lattices (2006, background): “The possibility to apply [50] to irreducible cocompact lattices enables us to prove the following.”
Isometry groups of non‐positively curved spaces: structure theory (2008, methodology): “Having established that N acts minimally and without a fixed point at infinity, we can apply the splitting theorem (Corollary 10 in [49]) and deduce from the irreducibility of X that N does not split.”