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It is well known that a complete Riemannian manifold M which is locally isometric to a symmetric space is covered by a symmetric space. Here, we prove that a discrete version of this property (called local to global rigidity) holds for a large class of vertex-transitive graphs, including Cayley graphs of torsion-free lattices in simple Lie groups, and Cayley graphs of torsion-free virtually nilpotent groups. By contrast, we exhibit various examples of Cayley graphs of finitely presented groups (for example, SL4(Z)) which fail to have this property, answering a question of Benjamini and Georgakopoulos. Answering a question of Cornulier, we also construct a continuum of pairwise nonisometric large-scale simply connected locally finite vertex-transitive graphs. This question was motivated by the fact that large-scale simply connected Cayley graphs are precisely Cayley graphs of finitely presented groups and therefore have countably many isometric classes. [ABSTRACT FROM AUTHOR]

We prove that connected higher-rank simple Lie groups have Lafforgue's strong property (T) with respect to a certain class of Banach spaces 10 containing many classical superreflexive spaces and some non-reflexive spaces as well. This generalizes the result of Lafforgue asserting that SL(3,R) has strong property (T) with respect to Hilbert spaces and the more recent result of the second-named author asserting that SL(3,R) has strong property (T) with respect to a certain larger class of Banach spaces. For the generalization to higher-rank groups, it is sufficient to prove strong property (T) for Sp(2,R) and its universal covering group. As consequences of our main result, it follows that for X 10, connected higher-rank simple Lie groups and their lattices have property (FX) of Bader, Furman, Gelander and Monod, and that the expanders constructed from a lattice in a connected higher-rank simple Lie group do not admit a coarse embedding into X. [ABSTRACT FROM AUTHOR]