Consider a finitely presented group \(\Gamma\) and a reductive affine algebraic group \(G\) over \(\mathbb C\). This article studies normal coverings of the \(G\)-character variety of \(\Gamma\), \(\chi_\Gamma(G):=\text{hom}(\Gamma,G)//G\) (algebraic geometric quotient under conjugation), when the fundamental group \(\pi_1(G)\) has no torsion.
The first main result proves that any covering homomorphism of such groups \(p:H\mapsto G\) induces a covering between corresponding character varieties \(p_*:\chi_\Gamma(G)\mapsto \chi_\Gamma(H)\), with fiber \(\chi_\Gamma(\text{ker }p)\). The same statement is valid in the compact Lie group category, i.e., replacing \(G\) by a maximal compact subgroup.
Let now \(\Sigma\) be a (real) surface of genus \(g\) with \(p\) punctures. The second main result is the formula
\[ \pi_1(\chi_{\pi_1(\Sigma)}(G))=\pi_1(G)^{b_1(\Sigma)} \] where \(b_1(\Sigma)\) is the first Betti number of \(X\), so it equals \(2g\) in the closed surface case (no punctures) and \(2g+p-1\) in the open (punctured) surface case. The formula above also works, when \(\Sigma\) is replaced by an \(n\)-dimensional (real) torus (replacing \(b_1(\Sigma)\) by \(n\)), in the cases when the derived group of \(G\) is a product of simple groups of type \(A_n\) or \(C_n\). It has also been recently extended to the case where \(\pi_1(G)\) has torsion in [I. Biswas, S. Lawton and D. A. Ramras, Math. Z. 281, No. 1-2, 415–425 (2015; Zbl 1349.14042)].
Let now \(\Gamma^g\) be the fundamental group of a closed surface \(\Sigma_g\) of genus \(g\geq 1\). Note that, under the non-Abelian Hodge correspondence, \(\chi_{\Gamma^g}(G)\) is homeomorphic to a certain moduli space of (semistable) \(G\)-Higgs bundles over \(\Sigma_g\). Thus, the above formula can be seen as an extension of the formula, obtained in [S. B. Bradlow, O. García-Prada and P. B. Gothen, Topology 47, No. 4, 203–224 (2008; Zbl 1165.14028)], for the moduli space of \(GL_n(\mathbb C)\)-Higgs bundles with coprime degree and rank (the smooth case) over \(\Sigma_g\).
The last part of the article presents an interesting application of these results to the stable character varieties of surface groups, defined as a natural colimit \(\chi_{\Gamma^g}(SU):=\text{colim}_{n\mapsto\infty}\chi_{\Gamma^g}(SU(n))\). Note that by the Narasimhan-Seshadri Theorem, there is a homeomorphism between the character variety \(\chi_{\Gamma^g}(SU(n))\) (the compact version of \(\chi_{\Gamma^g}(SL_n(\mathbb C))\)) and the moduli space of (semistable) trivial determinant vector bundles over \(\Sigma_g\). The last main result is the proof of the homotopy equivalence between the stable character varieties \(\chi_{\Gamma^g}(SU)\) and \(\mathbb CP^\infty\) (proving that their homotopy type is independent of the genus).
This result has a very nice connection with geometric quantization of the moduli space of bundles: it is well known that the moduli spaces of bundles \(\chi_{\Gamma^g}(SU(n))\) carry a natural so-called determinant line bundle (which coincides with the pre-quantum line bundle for \(\chi_{\Gamma^g}(SU(n))\) in the context of geometric quantization) whose holomorphic sections form the vector space of conformal blocks of a conformal field theory, the dimension of which was computed by the celebrated Verlinde formula. Since \(BU(1)\), the classifying space for \(\mathbb C\)-line bundles, is just \(\mathbb CP^\infty\), a map \(\chi_{\Gamma^g}(SU)\mapsto\mathbb CP^\infty\) defines a line bundle by pull-back of the universal line bundle over \(\mathbb CP^\infty\). Following a question stated in this article, it has been shown in the recent preprint [L. C. Jeffrey, D. A. Ramras and J. Weitsman, “The prequantum line bundle on the moduli space of flat \(SU(n)\) connections on a Riemann surface and the homotopy of the large \(n\) limit”, arXiv:1411.4360] that the map \(\chi_{\Gamma^g}(SU)\mapsto \mathbb CP^\infty\) corresponding to the determinant line bundle over \(\chi_{\Gamma^g}(SU)\) (which behaves well under stabilization in the large \(n\) limit) is indeed a homotopy equivalence.
Finally, in the Appendix, written by N.-K. Ho and C.-C. M. Liu, it is shown that, for \(\Sigma_g\) a hyperbolic surface \((g\geq 2)\) and a complex reductive group \(G\), \(\pi_0(\text{Hom}(\pi_1(\Sigma_g)))\cong \pi_1(DG)\), where \(DG\) is the derived group of \(G\), generalizing the same statement previously established in the complex semisimple case.