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WikiProject Mathematics (Rated Start-class, Low-priority)
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 Field:  Geometry

Arithmetic lattice

If Lambda is an arithmetic lattice in, say, SO(3,5) x SO(8) that has dense projection into SO(8) and has trivial intersection with {I} x SO(8), then the image Gamma of Lambda in SO(3,5) is a lattice in SO(3,5). The inverse map gives a representation of Gamma into SO(3,5) x SO(8) that does not extend to SO(3,5). Here, G = SO(3,5) and F = \R. So I *think* that some adjustment needs to be made in how the result is stated in the article, unfortunately. Maybe just put in a hypothesis that the image of the representation should have simple Zariski closure? Possibly I'm misreading something here?

I also wonder whether we need to have a qualification in the statement that we have first to restrict the representation to a subgroup of finite index before it will extend to the ambient Lie group.

I've noticed many times that it's very tricky to state superrigidity without having so many technical hypotheses that the usefulness of the result is made unclear. :-(

Scot Adams

If I remember correctly one of the hypotheses in Margulis' superrigidity theorem is that the image of the lattice be unbounded. As you note this is essential since Galois embeddings of the field of definition of the lattice give morphisms which do not extend in general. This is also essential for the proof of arithmeticity since the fact that images at finite places are bounded give you integrality. If that's OK with everybody I will add the boundedness condition and a precise reference to Margulis' book for the statement. This condition was mentioned in the statement so I think it is correct. I added the reference to Margulis' book. jraimbau (talk) 13:17, 17 March 2017 (UTC)
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