Density of Zariski density for surface groups (Q2510817)

If \(\Gamma\) is the fundamental group of a closed surface and \(G\) is a real Lie group so that \(\text{Hom}(\Gamma,G)\) has a natural structure of an analytic space, then a homomorphism \(\varphi:\Gamma\to G\) is: (i)\, smooth if \(\text{Hom}(\Gamma,G)\) is a smooth analytic manifold near \(\varphi\) and its dimension equals its virtual dimension \(\text{vdim}(\text{Hom}(\Gamma,G))\), (ii)\, fully flexible if smooth homomorphisms are dense in a neighborhood of \(\varphi\) in \(\text{Hom}(\Gamma,G)\), (iii)\, partially rigid if there is a neighborhood of \(\varphi\) in \(\text{Hom}(\Gamma,G)\) which contains no smooth homomorphisms. In this paper, the authors consider problems of existence of homomorphisms \(\Gamma\to G\) which are partially rigid, fully flexible, or neither. They show that if \(G\) is a connected reductive real algebraic group and \(\Gamma\) is the fundamental group of a closed surface of genus greater than \(1\), then the trivial homomorphism \(\Gamma\to G\) can be deformed into fully flexible homomorphisms, and if the genus is greater than \(2\dim(G)^2\), then homomorphisms \(\Gamma\to G\) are either fully flexible or partially rigid. Also, the authors consider surface groups in real reductive algebraic groups which are fully flexible. They show that if \(G\) is a semisimple real algebraic group, \(\Gamma\) is the fundamental group of a closed surface of genus greater than \(2\dim(G)^2\), and \(\varphi:\Gamma\to G\) is a homomorphism with reductive Zariski closure, then \(\varphi\) is fully flexible if and only if the center \(c\) of the centralizer of \(\varphi(\Gamma)\) in \(\mathfrak{g}\) is balanced with respect to \(\varphi\). And if there exists a non-fully flexible homomorphism \(\varphi:\Gamma\to G\), then the Zariski closure of \(\varphi(\Gamma)\) admits a transitive isometric action on a tube type Hermitian symmetric space, and the action of \(\Gamma\) on this space is a maximal representation.