Periodic Reporting for period 1 - HORUS (THE NON-ABELIAN HODGE THEORY OF AN ORBIFOLD KLEIN SURFACE)
Berichtszeitraum: 2018-09-01 bis 2020-08-31
As developed by N. Hitchin, F. Labourie, V. Fock, A. Goncharov, O. Guichard and A. Wienhard, Higher Teichmüller Theory makes sense for fundamental groups of orientable surfaces. One specific goal of the theory is to construct discrete and faithful representations of such fundamental groups and to understand the topology of the Higher Teichmüller Spaces thus constructed. The main examples of Higher Teichmüller Spaces are Hitchin components and spaces of maximal representations, and a conjecture of Guichard, Labourie and Wienhard suggests that positive representations are also discrete and faithful.
The main goal of the HORUS project was to develop Higher Teichmüller Theory for orbifold fundamental groups. Indeed, the classical Teichmüller theory of such groups is well understood, but their higher rank analogues have only been studied in rank 2, by S. Choi and W. Goldman. In collaboration with Daniele Alessandrini and Gye-Seon Lee, we studied orbifold Hitchin components in arbitrary rank. To do so, we extended Hitchin's Higgs bundles techniques to the orbifold case, in order to obtain an analytic parameterization of the Hitchin components in this case too.
As an application, we constructed new examples of discrete and faithful projective representations of hyperbolic Coxeter groups. We also used our techniques to study the rigidity properties of real projective 3-manifolds.
The main goal of the HORUS project was to develop Higher Teichmüller Theory for orbifold fundamental groups. Indeed, the classical Teichmüller theory of such groups is well understood, but their higher rank analogues have only been studied in rank 2, by S. Choi and W. Goldman. In collaboration with Daniele Alessandrini and Gye-Seon Lee, we studied orbifold Hitchin components in arbitrary rank. To do so, we extended Hitchin's Higgs bundles techniques to the orbifold case, in order to obtain an analytic parameterization of the Hitchin components in this case too.
As an application, we constructed new examples of discrete and faithful projective representations of hyperbolic Coxeter groups. We also used our techniques to study the rigidity properties of real projective 3-manifolds.
The main result of the joint work with D. Alessandrini and G.S. Lee is a parameterization of Hitchin components of orbifold fundamental groups in terms of analytic differentials on compact orbi-curves of negative Euler characteristic. This generalizes Hitchin's result in the closed orientable case. From this we deduce (1) that Hitchin components are contractible spaces and (2) a formula to compute the dimension of these spaces. This result has been accepted for publication by the Journal of the European Mathematical Society (JEMS). We have already given several applications of the main result, which has been presented in various scientific meetings, including the CIRM congress on Teichmüller Theory in Luminy (october 2020).
Additionally, research on related topics has been pursued in collaboration with other colleagues. With Victoria Hoskins, we have studied group actions on moduli spaces of quiver representations and we have in particular applied this to the study of rational points of quiver varieties. And with Indranil Biswas, we have started to study parabolic vector bundles on Klein surfaces. These results have been accepted for publication, respectively in the International Journal of Mathematics, the Annales de l'Institut Fourier and the Illinois Journal of Mathematics.
Additionally, research on related topics has been pursued in collaboration with other colleagues. With Victoria Hoskins, we have studied group actions on moduli spaces of quiver representations and we have in particular applied this to the study of rational points of quiver varieties. And with Indranil Biswas, we have started to study parabolic vector bundles on Klein surfaces. These results have been accepted for publication, respectively in the International Journal of Mathematics, the Annales de l'Institut Fourier and the Illinois Journal of Mathematics.
Before this project, Hitchin components for orbifold fundamental groups had only been studied for the groups PGL(2,R) and PGL(3,R). In contrast, the new results obtained over the course of the project hold for an infinite family of groups: the groups PGL(n,R) for all n, and, more generally, all split real forms of simple connected complex Lie groups with trivial center. As a follow-up, we intend to study maximal representations of orbifold fundamental groups. Orbifold fundamental groups also constitute a prime source of examples to construct twisted character varieties. By analyzing such examples carefully, we expect to gain a better understanding of the nonabelian Hodge correspondence with nonconstant coefficients, which is a promising lead for future research.