The topology of moduli spaces of free group representations

Abstract

For any complex affine reductive group G and a fixed choice of maximal compact subgroup K, we show that the G-character variety of a free group strongly deformation retracts to the corresponding K-character space, which is a real semi-algebraic set. Combining this with constructive invariant theory and classical topological methods, we show that the SL(3,C)-character variety of a rank 2 free group is homotopic to an 8 sphere and the SL(2,C)-character variety of a rank 3 free group is homotopic to a 6 sphere.

arXiv comment37 pages, 2 figures, version 2 corrects typos, generalizes context, and adds a corollary, it is the accepted version to appear in Mathematische Annalen

References in corpus (5)

• Generators, Relations and Symmetries in Pairs of 3x3 Unimodular Matrices
• Homotopy groups of moduli spaces of representations
• Poisson Geometry of SL(3,C)-Character Varieties Relative to a Surface with Boundary
• Quantizations of Character Varieties and Quantum Knot Invariants
• Invariants of 2 by 2 matrices, irreducible SL(2,C) characters and the Magnus trace map

Cited by in corpus (31)

• Topology of character varieties of Abelian groups
• Singularities of free group character varieties
• The topology of nilpotent representations in reductive groups and their maximal compact subgroups
• Locally homogeneous geometric manifolds
• E-polynomial of the SL(3,C)-character variety of free groups
• Homotopy Groups of Free Group Character Varieties
• Topology of Moduli Spaces of Free Group Representations in Real Reductive Groups
• Fundamental Groups of Character Varieties: Surfaces and Tori
• Covering spaces of character varieties
• Serre polynomials of $S{L}_n$- and $PG{L}_n$-character varieties of free groups
• E-polynomial of SL(2,C)-Character Varieties of Free groups
• Character varieties for real forms
• Invariants of pairs in SL(4,C) and SU(3,1)
• Flawed groups and the topology of character varieties
• Character varieties of virtually nilpotent Kähler groups and G-Higgs bundles
• Character Varieties and the Moduli of Quiver Representations
• Representation varieties of twisted Hopf links
• The Topology of Parabolic Character Varieties of Free Groups
• Special elliptic isometries, relative SU(2,1)-character varieties, and bendings
• Geometry of $\mathrm{SU}(3)$-character varieties of torus knots
• Poisson maps between character varieties: gluing and capping
• Homotopy Type of Moduli Spaces of G-Higgs Bundles and Reducibility of the Nilpotent Cone
• Yang-Mills-Higgs connections on Calabi-Yau manifolds, II
• Fundamental Group of Moduli Spaces of Representations
• Character Varieties of Generalized Torus Knot Groups
• Non-ergodicity on SU(2) and SU(3) character varieties of the once-punctured torus
• Hodge-Deligne Polynomials of Symmetric Products of Algebraic Groups
• $e$-polynomials of character varieties
• Dynamics on the SU(2,1)-character variety of the one-holed torus
• Newton-Okounkov polyhedra for character varieties and configuration spaces
• On the types of the mixed Hodge structures of character varieties