Topology of Moduli Spaces of Free Group Representations in Real Reductive Groups

Let $G$ be a real reductive algebraic group with maximal compact subgroup $K$, and let $F_r$ be a rank $r$ free group. We show that the space of closed orbits in $\mathrm{Hom}(F_r,G)/G$ admits a strong deformation retraction to the orbit space $\mathrm{Hom}(F_r,K)/K$. In particular, all such spaces have the same homotopy type. We compute the Poincar\'e polynomials of these spaces for some low rank groups $G$, such as $\mathrm{Sp}(4,\mathbb{R})$ and $\mathrm{U}(2,2)$. We also compare these real moduli spaces to the real points of the corresponding complex moduli spaces, and describe the geometry of many examples.Comment: v2: exposition improved, typos corrected, and a minor gap in a proof fixed; 25 pages; accepted at Forum Mathematicu

Topology of moduli spaces of free group representations in real reductive groups

This work was partially supported by the projects PTDC/MAT-GEO/0675/2012 and PTDC/MAT/120411/2010, FCT, Portugal. The authors also acknowledge support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 "RNMS: Geometric structures and Representation varieties" (the GEAR Network). Additionally, the third author was partially supported by the Simons Foundation grant 245642 and the U.S. National Science Foundation grant DMS 1309376, and the fourth author was partially supported by Centro de Matematica da Universidade de Tras-os-Montes e Alto Douro (PEst-OE/MAT/UI4080/2011).Let G be a real reductive algebraic group with maximal compact subgroup K, and let F-r be a rank r free group. We show that the space of closed orbits in Hom(F-r, G)/G admits a strong deformation retraction to the orbit space Hom(F-r, K)/K. In particular, all such spaces have the same homotopy type. We compute the Poincare polynomials of these spaces for some low rank groups G, such as Sp(4, IR) and U(2, 2). We also compare these real moduli spaces to the real points of the corresponding complex moduli spaces, and describe the geometry of many examples.publishersversionpublishe

Singularities of free group character varieties

Let X be the moduli space of SL(n,C), SU(n), GL(n,C), or U(n)-valued representations of a rank r free group. We classify the algebraic singular stratification of X. This comes down to showing that the singular locus corresponds exactly to reducible representations if there exist singularities at all. Then by relating algebraic singularities to topological singularities, we show the moduli spaces X generally are not topological manifolds, except for a few examples we explicitly describe.Comment: 33 pages. Version 4 is shorter and more focused; cut material will be expanded upon and written up in subsequent papers. Clarifications, and expository revisions have been added. Accepted for publication in Pacific Journal of Mathematic

Fundamental Groups of Character Varieties: Surfaces and Tori

We compute the fundamental group of moduli spaces of Lie group valued representations of surface and torus groups.Comment: v2: 12 pages, minor edits, accepted for publication at Mathematische Zeitschrif

Connectedness of Higgs bundle moduli for complex reductive Lie groups

We carry an intrinsic approach to the study of the connectedness of the moduli space $\mathcal{M}_G$ of $G$-Higgs bundles, over a compact Riemann surface, when $G$ is a complex reductive (not necessarily connected) Lie group. We prove that the number of connected components of $\mathcal{M}_G$ is indexed by the corresponding topological invariants. In particular, this gives an alternative proof of the counting by J. Li of the number of connected components of the moduli space of flat $G$-connections in the case in which $G$ is connected and semisimple.Comment: Due to some mistake the authors did not appear in the previous version. Fixed this. Final version; to appear in the Asian Journal of Mathematics. 19 page

Higgs bundles and higher Teichm\"uller spaces

This paper is a survey on the role of Higgs bundle theory in the study of higher Teichm\"uller spaces. Recall that the Teichm\"uller space of a compact surface can be identified with a certain connected component of the moduli space of representations of the fundamental group of the surface into $\mathrm{PSL}(2,{\mathbb{R}})$. Higher Teichm\"uller spaces correspond to special components of the moduli space of representations when one replaces $\mathrm{PSL}(2,{\mathbb{R}})$ by a real non-compact semisimple Lie group of higher rank. Examples of these spaces are provided by the Hitchin components for split real groups, and maximal Toledo invariant components for groups of Hermitian type. More recently, the existence of such components has been proved for $\mathrm{SO}(p,q)$, in agreement with the conjecture of Guichard and Wienhard relating the existence of higher Teichm\"uller spaces to a certain notion of positivity on a Lie group that they have introduced. We review these three different situations, and end up explaining briefly the conjectural general picture from the point of view of Higgs bundle theory.Comment: arXiv admin note: substantial text overlap with arXiv:1511.0775