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

Abstract

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 real rank 2 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 Teichmüller 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 of 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.The spaces X max,sw2 sw1 (Γ, PSp(4, R)) are singular, but the singularities consist only of Z 2 and Z 2 ⊕ Z 2 -orbifold points. The space H ′ /Z 2 also has an orbifold structure. The homeomorphism above is an orbifold isomorphism, in particular, it is smooth away from the singular set.Corollary 7. Each space X max,sw2 sw1 (Γ, PSp(4, R)) deformation retracts onto the quotient of (S 1 ) 2g−2 by inversion. In particular, its rational cohomology is: PSp(4, R)), Q) ∼ = H j ((S 1 ) 2g−2 , Q) if j is even, 0 otherwise. Character varieties and Higgs bundlesIn this section we recall general facts about character varieties and Higgs bundles.

The moduli spaces of equivariant minimal surfaces in $${\mathbb {RH}}^3$$ and $$\mathbb {RH}^4$$ via Higgs bundles

“…When G = Sp(4, R) the Hitchin representations are all maximal (i.e., have maximal Toledo invariant) and Collier [11] extended Labourie's result to cover all maximal representations in Sp (4, R). Very recently Collier and collaborators have further extended this uniqueness result to maximal representations in P Sp(4, R) [1] and SO (2, n) [12]. Since P Sp (4, R) SO 0 (2, 3) an adaptation of the techniques of this current paper may shed some light on the minimal surfaces for non-maximal representations.…”

“…(2.6)). Thus we can write the holomorphic structure for V in the form∂ 1) and the holomorphic structure depends only upon…”

Higgs Bundles and Geometric Structures on Manifolds

“…Higgs bundles are an important tool in higher Teichmüller theory because they can be used to describe the topology of the character varieties (see for example Hitchin [26,27], Alessandrini-Collier [1]). Anyway, they were initially believed to give very little information on the geometry of a single representation.…”

“…This is another way to generalize Fuchsian representations to higher rank Lie groups. For G = Sp(4, R) and PSp(4, R), an explicit description of the topology of the maximal components was determined in a joint work with Brian Collier [1].…”

“…For example, in the character variety of X (π 1 (S), PGL(2, C)) we have the open subset of quasi-Fuchsian representations, denoted by QFuch(S). Quasi-Fuchsian representations can be defined as those representations whose action on CP 1 is topologically conjugate to the action of a Fuchsian representation on CP 1 . The open subset QFuch(S) is homeomorphic to R 12g−12 .…”