Surface Group Representations Research Articles

The geometry of maximal components of the PSp(4, ℝ) character variety

In this paper we describe the space of maximal components of the character variety of surface group representations into PSp(4,R) and Sp(4,R). For every rank 2 real Lie group of Hermitian type, we construct a mapping class group invariant complex structure on the maximal components. For the groups PSp(4,R) and Sp(4,R), we give a mapping class group invariant parameterization of each maximal component as an explicit holomorphic fiber bundle over Teichm\"uller space. Special attention is put on the connected components which are singular, we give a precise local description of the singularities and their geometric interpretation. We also describe the quotient of the maximal components for PSp(4,R) and Sp(4,R) by the action of the mapping class group as a holomorphic submersion over the moduli space of curves. These results are proven in two steps, first we use Higgs bundles to give a non-mapping class group equivariant parameterization, then we prove an analogue of Labourie's conjecture for maximal PSp(4,R) representations.

Representations of surface groups in the projective general linear group

Given a closed, oriented surface X of genus g>1, and a semisimple Lie group G, let R_G be the moduli space of reductive representations of the fundamental group of X in G. We determine the number of connected components of R_PGL(n,R), for n>=4 even. In order to have a first division of connected components, we first classify real projective bundles over such a surface. Then we achieve our goal, using holomorphic methods through the theory of Higgs bundles over compact Riemann surfaces. We also show that the complement of the Hitchin component in R_SL(3,R) is homotopically equivalent to R_SO(3).

Brackets in representation algebras of Hopf algebras

For any graded bialgebras A and B , we define a commutative graded algebra A_B representing the functor of B -representations of A . When A is a cocommutative graded Hopf algebra and B is a commutative ungraded Hopf algebra, we introduce a method deriving a Gerstenhaber bracket in A_B from a Fox pairing in A and a balanced biderivation in B . Our construction is inspired by Van den Bergh’s non-commutative Poisson geometry, and may be viewed as an algebraic generalization of the Atiyah–Bott–Goldman Poisson structures on moduli spaces of representations of surface groups.

Exotic components of SO(p,q) surface group representations, and their Higgs bundle avatars

For semisimple Lie groups, moduli spaces of Higgs bundles on a Riemann surface correspond to representation varieties for the surface fundamental group. In many cases, natural topological invariants label connected components of the moduli spaces. Hitchin representations into split real forms, and maximal representations into Hermitian Lie groups, are the only previously know cases where natural invariants do not fully distinguish connected components. In this note we announce the existence of new such exotic components in the moduli spaces for the groups SO(p,q) with 2<p<q. These groups lie outside formerly know classes of groups associated with exotic components.

Quaternionic hyperbolic Fenchel–Nielsen coordinates

Let $$\mathrm {Sp}(2,1)$$ be the isometry group of the quaternionic hyperbolic plane $$\mathbf{H}_{\mathbb {H}}^2$$ . An element g in $$\mathrm {Sp}(2,1)$$ is hyperbolic if it fixes exactly two points on the boundary of $$\mathbf{H}_{\mathbb {H}}^2$$ . We classify pairs of hyperbolic elements in $$\mathrm {Sp}(2,1)$$ up to conjugation. A hyperbolic element of $$\mathrm {Sp}(2,1)$$ is called loxodromic if it has no real eigenvalue. We show that the set of $$\mathrm {Sp}(2,1)$$ conjugation orbits of irreducible loxodromic pairs is a $$(\mathbb {C}\mathbb {P}^1)^4$$ bundle over a topological space that is locally a semi-analytic subspace of $$\mathbb {R}^{13}$$ . We use the above classification to show that conjugation orbits of ‘geometric’ representations of a closed surface group (of genus $$g \ge 2$$ ) into $$\mathrm {Sp}(2,1)$$ can be determined by a system of $$42g-42$$ real parameters. Further, we consider the groups $$\mathrm {Sp}(1,1)$$ and $$\mathrm {GL}(2, \mathbb {H})$$ . These groups also act by the orientation-preserving isometries of the four and five dimensional real hyperbolic spaces respectively. We classify conjugation orbits of pairs of hyperbolic elements in these groups. These classifications determine conjugation orbits of ‘geometric’ surface group representations into these groups.

Pappus Theorem, Schwartz Representations and Anosov Representations

In the paper Pappus's theorem and the modular group [13], R. Schwartz constructed a 2-dimensional family of faithful representations ρΘ of the modular group PSL(2, Z) into the group G of projective symmetries of the projective plane via Pappus Theorem. If PSL(2, Z)o denotes the unique index 2 subgroup of PSL(2, Z) and PGL(3, R) the subgroup of G consisting of projective transformations, then the image of PSL(2, Z)o under ρΘ is in PGL(3, R). The representations ρΘ share a very interesting property with Anosov representations of surface groups into PGL(3, R): It preserves a topological circle in the flag variety. However, the representation ρΘ itself cannot be Anosov since the Gromov boundary of PSL(2, Z) is a Cantor set and not a circle. In her PhD Thesis [15], V. P. Valerio elucidated the Anosov-like feature of the Schwartz representations by showing that for each representation ρΘ, there exists an 1-dimensional family of representations (ρ e Θ) e∈R of PSL(2, Z)o into PGL(3, R) such that ρ 0 Θ is the restriction of the Schwartz representation ρΘ to PSL(2, Z)o and ρ e Θ is Anosov for every e < 0. This result was announced and presented in her paper [14]. In the present paper, we extend and improve her work. For every representation ρΘ, we build a 2-dimensional family of representations (ρ λ Θ) λ∈R 2 of PSL(2, Z)o into PGL(3, R) such that ρ λ Θ = ρ e Θ for λ = (e, 0) and ρ λ Θ is Anosov for every λ ∈ R • , where R • is an open set of R 2 containing {(e, 0) | e < 0}. Moreover, among the 2-dimensional family of new Anosov representations, an 1-dimensional subfamily of representations can extend to representations of PSL(2, Z) into G, and therefore the Schwartz representations are, in a sense, on the boundary of the Anosov representations in the space of all representations of PSL(2, Z) into G.

Torsion and symplectic volume in Seifert manifolds

For any oriented Seifert manifold X and compact connected Lie group G with finite center, we relate the Reidemeister density of the moduli space of representations of the fundamental group of X into G to the Liouville measure of some moduli spaces of representations of surface groups into G.

Characteristic Classes of Fiberwise Branched Surface Bundles via Arithmetic Groups

This paper is about the cohomology of certain finite-index subgroups of mapping class groups and its relation to the cohomology of arithmetic groups. For G=Z/mZ and for a regular G-cover S→S¯ (possibly branched), a finite-index subgroup Γ<Mod(S¯) acts on H1(S;Z) commuting with the deck group action, thus inducing a homomorphism Γ→Sp2gG(Z) to an arithmetic group. The induced map H∗(Sp2gG(Z);Q)→H∗(Γ;Q) can be understood using index theory. To this end, we describe a families version of the G-index theorem for the signature operator and apply this to (i) compute H2(Sp2gG(Z);Q)→H2(Γ;Q), (ii) rederive Hirzebruch’s formula for signature of a branched cover, (iii) compute Toledo invariants of surface group representations to SU(p,q) arising from Atiyah–Kodaira constructions, and (iv) describe how classes in H∗(Sp2gG(Z);Q) give equivariant cobordism invariants for surface bundles with a fiberwise G action, following Church–Farb–Thibault.

Incompressible solvable representations of surface groups

The fundamental group of every surface that is not the projective plane or Klein bottle has a representation to a torsion-free group of upper-triangular matrices in SL2(R) with no simple loop (i.e. a nontrivial element representing a simple closed curve) in the kernel.

On weakly maximal representations of surface groups

We introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called weakly maximal representations. We prove that weakly maximal representations are discrete and injective and we describe the structure of the Zariski closure of their image. Furthermore, we prove that the set of weakly maximal representations is a closed subset of the representation variety and describe its relation to other geometrically significant subsets of the representations variety.

On Fenchel–Nielsen Coordinates of Surface Group Representations into SU(3,1)

AbstractLet Σgbe a compact, connected, orientable surface of genusg≥ 2. We ask for a parametrisation of the discrete, faithful, totally loxodromic representations in the deformation space Hom(π1(Σg), SU(3, 1))/SU(3, 1). We show that such a representation, under some hypothesis, can be determined by 30g− 30 real parameters.

Dominating surface group representations and deforming closed anti-de Sitter 3–manifolds

In a previous paper by Deroin-Tholozan, the authors construct a map $\mathbf{\Psi}_\rho$ from the Teichm\"uller space of $S$ to itself and prove that, when $M$ has sectional curvature $\leq -1$, the image of $\mathbf{\Psi}_\rho$ lies (almost always) in the domain $\mathrm{Dom}(\rho)$ of Fuchsian representations stricly dominating $\rho$. Here we prove that $\mathbf{\Psi}_\rho: \mathrm{Teich}(S) \to \mathrm{Dom}(\rho)$ is a homeomorphism. As a consequence, we are able to describe the topology of the deformation space of anti-de Sitter structures on closed 3-manifolds.

Surface group representations to SL(2, ℂ) and Higgs bundles with smooth spectral data

We show that for every nonelementary representation of a surface group into $SL(2,{\mathbb C})$ there is a Riemann surface structure such that the Higgs bundle associated to the representation lies outside the discriminant locus of the Hitchin fibration.

Conical limit points and the Cannon-Thurston map

Let G G be a non-elementary word-hyperbolic group acting as a convergence group on a compact metrizable space Z Z so that there exists a continuous G G -equivariant map i : ∂ G → Z i:\partial G\to Z , which we call a Cannon-Thurston map. We obtain two characterizations (a dynamical one and a geometric one) of conical limit points in Z Z in terms of their pre-images under the Cannon-Thurston map i i . As an application we prove, under the extra assumption that the action of G G on Z Z has no accidental parabolics, that if the map i i is not injective, then there exists a non-conical limit point z ∈ Z z\in Z with | i − 1 ( z ) | = 1 |i^{-1}(z)|=1 . This result applies to most natural contexts where the Cannon-Thurston map is known to exist, including subgroups of word-hyperbolic groups and Kleinian representations of surface groups. As another application, we prove that if G G is a non-elementary torsion-free word-hyperbolic group, then there exists x ∈ ∂ G x\in \partial G such that x x is not a “controlled concentration point” for the action of G G on ∂ G \partial G .

Dominating Surface Group Representations by Fuchsian Ones

We prove that a representation from the fundamental group of a closed surface of negative Euler characteristic with values in the isometry group of a Riemannian manifold of sectional curvature bounded by −1 can be dominated by a Fuchsian representation. Moreover, we prove that the domination can be made strict, unless the representation is discrete and faithful in restriction to an invariant totally geodesic 2-plane of curvature −1. When applied to representations into PSL(2,R) of non-extremal Euler class, our result is a step forward in understanding the space of closed anti-de Sitter 3-manifolds.

Lyapunov Exponents for Surface Group Representations

Let (ρλ)λ∈Λ be a holomorphic family of representations of a surface group π1(S) into PSL(2,C), where S is a topological (possibly punctured) surface with negative Euler characteristic. Given a structure of Riemann surface of finite type on S we construct a bi- furcation current on the parameter space Λ, that is a (1,1) positive closed current attached to the bifurcations of the family. It is defined as the ddc of the Lyapunov exponent of the representation with respect to the Brownian motion on the Riemann surface S, endowed with its Poincar e metric. We show that this bifurcation current describes the asymptotic distribution of various codimension 1 phenomena in Λ. For instance, the random hypersur- faces of Λ defined by the condition that a random closed geodesic on S is mapped under ρλ to a parabolic element or the identity are asymptotically equidistributed with respect to the bifurcation current. The proofs are based on our previous work [DD1], and on a careful control of a discretization procedure of the Brownian motion.

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.

Variations along the Fuchsian locus

The main result is an explicit expression for the Pressure Metric on the Hitchin component of surface group representations into PSL(n,R) along the Fuchsian locus. The expression is in terms of a parametrization of the tangent space by holomorphic differentials, and it gives a precise relationship with the Petersson pairing. Along the way, variational formulas are established that generalize results from classical Teichmueller theory, such as Gardiner's formula, the relationship between length functions and Fenchel-Nielsen deformations, and variations of cross ratios.

Spaces of surface group representations

Let $$\Gamma _g$$ denote the fundamental group of a closed surface of genus $$g \ge 2$$ . We show that every geometric representation of $$\Gamma _g$$ into the group of orientation-preserving homeomorphisms of the circle is rigid, meaning that its deformations form a single semi-conjugacy class. As a consequence, we give a new lower bound on the number of topological components of the space of representations of $$\Gamma _g$$ into $${{\mathrm{Homeo}}}_+(S^1)$$ . Precisely, for each nontrivial divisor $$k$$ of $$2g-2$$ , there are at least $$|k|^{2g} + 1$$ components containing representations with Euler number $$\frac{2g-2}{k}$$ . Our methods apply to representations of surface groups into finite covers of $${{\mathrm{PSL}}}(2,\mathbb {R})$$ and into $${{\mathrm{Diff}}}_+(S^1)$$ as well, in which case we recover theorems of W. Goldman and J. Bowden. The key technique is an investigation of stability phenomena for rotation numbers of products of circle homeomorphisms using techniques of Calegari–Walker. This is a new approach to studying deformation classes of group actions on the circle, and may be of independent interest.

Higgs bundles for the non-compact dual of the special orthogonal group

Higgs bundles over a closed orientable surface can be defined for any real reductive Lie group $$G$$ . In this paper we examine the case $$G=\mathrm {SO}^*(2n)$$ . We describe a rigidity phenomenon encountered in the case of maximal Toledo invariant. Using this and Morse theory in the moduli space of Higgs bundles, we show that the moduli space is connected in this maximal Toledo case. The Morse theory also allows us to show connectedness when the Toledo invariant is zero. The correspondence between Higgs bundles and surface group representations thus allows us to count the connected components with zero and maximal Toledo invariant in the moduli space of representations of the fundamental group of the surface in $$\mathrm {SO}^*(2n)$$ .