In the literature there are several definitions of "rigidity" (or "super-rigidity") of representations, adapted to the circumastances. I wonder what are the relations between them; I excuse in advance if the answers are known, after some research I could not find them. I have singled out a few definitions for concreteness.

Notation first: let $\Gamma = \pi_1(X, x_0)$ where $X$ is Kähler, and $G$ a (semi-)simple connected algebraic group.

  1. (classic rigidity): Every $\phi$ close to $\rho$ (in the analytic topology of $\text{Hom}(\Gamma, G)$) is conjugated to $\rho$ by an element of $G$.
  2. (super-rigidity, e.g. Margulis): The harmonic $\rho$-equivariant map $f \colon \tilde{X} \to G/K$ is totally geodesic.
  3. (Hermitian super-rigidity, e.g. Siu): Here we suppose $G/K$ is Hermitian symmetric. Then $f$ as above is holomorphic.
  4. (from Kim-Pansu, Duke Math. J., to appear): (Here we may need to suppose $X$ to be a surface of high enough genus) $\rho$ is rigid if every $\phi$ close enough to $\rho$ is not smooth in $\text{Hom}(\Gamma, G)$ (equivalently, under the assumption on $X$: not Zariski-dense).

The last one is a bit "ad hoc" to their situation, since it allows them to prove that $\text{Hom}(\Gamma, G)$ splits in connected components, either entirely rigid or where Zariski dense points form a dense subset.

I think that 3. has been introduced as an intermediate step to 2. (when $\tilde{X}$ is a Hermitian symmetric space), but I do not know how general this is. To me, the only clear implication is 1. $\implies$ 4. (if $\phi_n \to \rho$ are smooth, then $\dim T_{\phi_n}\text{Hom}(\Gamma, G) = vdim = (1-\chi(X)) \dim(G)$ but if a neighborhood of $\phi_n$ is conjugated to $\rho$ then $\dim T_{\phi_n}\text{Hom}(\Gamma, G) \leq \dim G$).

I also think that despite the "super-" name there is a good chance that 1. $\implies$ 3. (at least if $X$ is a surface); so, main question:

Is there any implication between 1. and 3.? Or counter-example?

More with general-knowledge purposes:

Is there any other general implication / counterexample of implications between the points above? For example, how generally 3. $\implies$ 2.? Is 4. much weaker then any other one?

Of course, I would also be happy with answers for $\Gamma$ a cocompact lattice in $H$ and $\tilde{X} = H/K'$, or even just $X$ a Riemann surface.

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.