# links and interactions between different approaches to (super-)rigidity

By super-rigidity I mean some theorems concerning the arithmetic subgroups in semi-simple Lie groups. According to Margulis "Discrete subgroups of semi-simple Lie groups" (the book published by Springer), the superrigidity can be formulated as :

Theorem: let $G$ be a connected semi-simple Lie group with finite center, real rank at least 2, without compact factors. Let $\Gamma$ be an irreducible lattice in $G$. Let $\psi:\Gamma\rightarrow H$ be a homomorphism into a real algebraic group $H$. Then the Zariski closure of the image $\psi(\Gamma)$ is a semi-simple subgroup. If moreover $\psi(\Gamma)$ is not relatively compact, that the Zariski closure of $\psi(\Gamma)$ has trivial center and does not have non-trivial compact factors, then $\psi$ extends uniquely to a continuous homomorphism from $G$ to $H$.

The approach of Margulis makes heavy use of ergodic theory. On the other hand another approach is available using mainly techniques from differential geometry, see N.Mok, Y.Siu, and S.Yeung, "geometric superrigidity" (Inv.math.113, pp.57-83) and its references. Roughly speaking, the study of arithmetic subgroups in semi-simple Lie groups is reduced to differo-geometric properties of the corresponding (locally) symmetric manifolds.

My question is what links can one find between the two approaches? Does it mean that certain ergodic structures in the theory of Margulis carry also a geometric nature so that they can be recovered using purely differential geometry of locally symmetric manifolds? Or, could one refine the geometric aspects by ergodic-theoretic results and ideas?

I'm only a beginner to the geometry of locally symmetric manifolds, and I know almost nothing about the interactions between the ergodic approach and the differo-geometric studies around arithmetic subgroups of Lie groups. Any references and comments are warmly welcomed.

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An amateur's question: the Ratner's theory about unipotent flows on certain homogeneous manifolds is related to Margulis' works on arithmetic groups. But is there any geometric ways to study the similar properties? Especially Ratner's theory describes explicitly the closure of orbits of unipotent flows, but is there differo-geometric approach to the question of "closure" proplem, as the notion of orbit closure is related to dynamic systems? Excuse me for the nonsense. – genshin Oct 27 '10 at 10:28
There's also a newer approach to superrigidity by Monod: ams.org/mathscinet-getitem?mr=2123025 ams.org/mathscinet-getitem?mr=2219304 – Ian Agol Oct 27 '10 at 15:29
There is no established connection between the two approaches, but there is a weak relation which goes as follows: Suppose that $X$ is the symmetric space of $G$, assume that $H$ is, say, a Lie group of noncompact type and $Y$ is its symmetric space. Geometrically, one looks at the equivariant harmonic maps $h:X\to Y$ and tries to show they are isometric. Since $h$ is harmonic, its admits a measurable equivariant extension to the ideal boundary of $X$, which is exactly the setup of Margulis' approach. From there on, the two approaches, however, diverge. – Misha Mar 10 '12 at 12:38