I heard that in complex hyperbolic space setting for example CH2, there is some deformation of lattice nontrivial. What confused me is it seems contradicting Mostow Rigidity. Could someone explain this?
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2$\begingroup$ Please state precisely what you heard. Most likely, it concerned infinite covolume isometry groups. Even in that case there is a lot of rigidity (due to Goldman-Millson and hence generalized in various ways, see e.g. arxiv.org/abs/0903.3706). $\endgroup$– Igor BelegradekCommented Jul 25, 2014 at 15:15
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$\begingroup$ actually the example i see is the deformation of scokotty group and the ICM talk of Schwatz. Does Strong rigidity rules out all the possibility of nontrivial deformation of lattice(Even in rank one)except PSL(2,R)? $\endgroup$– user42804Commented Jul 26, 2014 at 6:56
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1$\begingroup$ Your examples are of infinite covolume. Yes, Mostow rigidity (or even the Weil rigidity mentioned by Andy Sanders) rules out deformations of a lattice in the associated Lie group, but the lattice could a priori be deformed in a bigger Lie group. This is discussed in the paper I linked aboved. $\endgroup$– Igor BelegradekCommented Jul 26, 2014 at 11:42
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It follows from Weil's rigidity theorem http://www.jstor.org/stable/1970212 that given any uniform lattice in $SU(n,1)$ with $n>1,$ it has no local deformations, in fact no non-trivial first order deformations. If $n=1$ then the the symmetric space is the real hyperbolic plane and there is a large family of deformations for surface groups.
Now, if you don't require the lattice condition, then there may or may not be deformations. The case of a surface group is the most well understood (to me at least). It is a theorem of Toledo that a discrete surface group (here surface means closed, oriented and genus at least two) in $SU(2,1)$ which stabilizes a totally geodesic holomorphic hyperbolic plane in $\mathbb{CH}^2$ is locally rigid. But, surfaces groups which stabilize a totally real lagrangian hyperbolic plane in $\mathbb{CH}^2$ admit many deformations, these are so-called complex hyperbolic quasi-Fuchsian groups and their deformations are still far from well understood. Maybe Misha Kapovich (who posts here and is an expert) might add some more interesting comments at some point.
answered Jul 25, 2014 at 15:30