Surface Group Representations Research Articles

Canonical representations of surface groups

Canonical representations of surface groups

McShane Identities for Higher Teichmüller Theory and the Goncharov–Shen Potential

We derive generalizations of McShane’s identity for higher ranked surface group representations by studying a family of mapping class group invariant functions introduced by Goncharov and Shen, which generalize the notion of horocycle lengths. In particular, we obtain McShane-type identities for finite-area cusped convex real projective surfaces by generalizing the Birman–Series geodesic scarcity theorem. More generally, we establish McShane-type identities for positive surface group representations with loxodromic boundary monodromy, as well as McShane-type inequalities for general rank positive representations with unipotent boundary monodromy. Our identities are systematically expressed in terms of projective invariants, and we study these invariants: we establish boundedness and Fuchsian rigidity results for triple and cross ratios. We apply our identities to derive the simple spectral discreteness of unipotent-bordered positive representations, collar lemmas, and generalizations of the Thurston metric.

Pentagon representations and complex projective structures on closed surfaces

We define a class of representations of the fundamental group of a closed surface of genus $2$ to $\mathrm{PSL}_2 (\mathbb C)$: the pentagon representations. We show that they are exactly the non-elementary $\mathrm{PSL}_2 (\mathbb C)$-representations of surface groups that do not admit a Schottky decomposition, i.e. a pants decomposition such that the restriction of the representation to each pair of pants is an isomorphism onto a Schottky group. In doing so, we exhibit a gap in the proof of Gallo, Kapovich and Marden that every non-elementary representation of a surface group to $\mathrm{PSL}_2 (\mathbb C)$ is the holonomy of a projective structure, possibly with one branched point of order $2$. We show that pentagon representations arise as such holonomies and repair their proof.

Dominating $$ \textrm{CAT}\left( -1 \right) $$ surface group representations by Fuchsian ones

Dominating $$ \textrm{CAT}\left( -1 \right) $$ surface group representations by Fuchsian ones

Virtual Classes of Representation Varieties of Upper Triangular Matrices via Topological Quantum Field Theories

In this paper, we use a geometric technique developed by Gonz\'alez-Prieto, Logares, Mu\~noz, and Newstead to study the $G$-representation variety of surface groups $\mathfrak{X}_G(\Sigma_g)$ of arbitrary genus for $G$ being the group of upper triangular matrices of fixed rank. Explicitly, we compute the virtual classes in the Grothendieck ring of varieties of the $G$-representation variety and the moduli space of $G$-representations of surface groups for $G$ being the group of complex upper triangular matrices of rank $2$, $3$, and $4$ via constructing a topological quantum field theory. Furthermore, we show that in the case of upper triangular matrices the character map from the moduli space of $G$-representations to the $G$-character variety is not an isomorphism.

Ergodicity of the mapping class group action on Deroin–Tholozan representations

This note investigates the dynamics of the mapping class group action on compact connected components of relative character varieties of surface group representations into PSL(2, $\mathbb{R}$), discovered by Deroin and Tholozan. We apply symplectic methods developed by Goldman and Xia to prove that the action is ergodic.

Surface group representations in SL2(ℂ) with finite mapping class orbits

Given an oriented surface of positive genus with finitely many punctures, we classify the finite orbits of the mapping class group action on the moduli space of semisimple complex special linear two dimensional representations of the fundamental group of the surface. For surfaces of genus at least two, such orbits correspond to homomorphisms with finite image. For genus one, they correspond to the finite or special dihedral representations. We also obtain an analogous result for bounded orbits in the moduli space.

Representations of surface groups with universally finite mapping class group orbit

Representations of surface groups with universally finite mapping class group orbit

Random Unitary Representations of Surface Groups I: Asymptotic Expansions

In this paper, we study random representations of fundamental groups of surfaces into special unitary groups. The random model we use is based on a symplectic form on moduli space due to Atiyah, Bott and Goldman. Let Sigma _{g} denote a topological surface of genus gge 2. We establish the existence of a large n asymptotic expansion, to any fixed order, for the expected value of the trace of any fixed element of pi _{1}(Sigma _{g}) under a random representation of pi _{1}(Sigma _{g}) into mathsf {SU}(n). Each such expected value involves a contribution from all irreducible representations of mathsf {SU}(n). The main technical contribution of the paper is effective analytic control of the entire contribution from irreducible representations outside finite sets of carefully chosen rational families of representations.

A collar lemma for partially hyperconvex surface group representations

We show that a collar lemma holds for Anosov representations of fundamental groups of surfaces into S L ( n , R ) SL(n,\mathbb {R}) that satisfy partial hyperconvexity properties inspired from Labourie’s work. This is the case for several open sets of Anosov representations not contained in higher rank Teichmüller spaces, as well as for Θ \Theta -positive representations into S O ( p , q ) SO(p,q) if p ≥ 4 p\geq 4 . We moreover show that ‘positivity properties’ known for Hitchin representations, such as being positively ratioed and having positive eigenvalue ratios, also hold for partially hyperconvex representations.

The rank two p-curvature conjecture on generic curves

We prove the p-curvature conjecture for rank two vector bundles with connection on generic curves. Our ingredients include new deformation techniques for vector bundles with vanishing p-curvatures on families of varieties, as well as topological criteria for finiteness of surface group representations in terms of simple loops and pairs of pants.

Asymptotics of quantum representations of surface groups

Asymptotics of quantum representations of surface groups

Geometry of compact complex manifolds associated to generalized quasi-Fuchsian representations

We study the topology and geometry of compact complex manifolds associated to Anosov representations of surface groups and other hyperbolic groups in a complex semisimple Lie group $G$. These manifolds are obtained as quotients of the domains of discontinuity in generalized flag varieties $G/P$ constructed by Kapovich-Leeb-Porti (arXiv:1306.3837), and in some cases by Guichard-Wienhard (arXiv:1108.0733). For $G$-Fuchsian representations and their Anosov deformations, where $G$ is simple, we compute the homology of the domains of discontinuity and of the quotient manifolds. For $G$-Fuchsian and $G$-quasi-Fuchsian representations in simple $G$ of rank at least two, we show that the quotient manifolds are not K\"{a}hler. We also describe the Picard groups of these quotient manifolds, compute the cohomology of line bundles on them, and show that for $G$ of sufficiently large rank these manifolds admit nonconstant meromorphic functions. In a final section, we apply our topological results to several explicit families of domains and derive closed formulas for topological invariants in some cases. We also show that the quotient manifold for a $G$-Fuchsian representation in $\mathrm{PSL}_3(\mathbb{C})$ is a fiber bundle over a surface, and we conjecture that this holds for all simple $G$.

Relations in bounded cohomology

We explain some interesting relations in the degree three bounded cohomology of surface groups. Specifically, we show that if two faithful Kleinian surface group representations are quasi-isometric, then their bounded fundamental classes are the same in bounded cohomology. This is novel in the setting that one end is degenerate, while the other end is geometrically finite. We also show that a difference of two singly degenerate classes with bounded geometry is boundedly cohomologous to a doubly degenerate class, which has a nice geometric interpretation. Finally, we explain that the above relations completely describe the linear dependences between the `geometric' bounded classes defined by the volume cocycle with bounded geometry. We obtain a mapping class group invariant Banach sub-space of the reduced degree three bounded cohomology with explicit topological generating set and describe all linear relations.

SO(n, n+1)-surface group representations and Higgs bundles

SO(n, n+1)-surface group representations and Higgs bundles

The geometry of maximal representations of surface groups into SO0(2,n)

In this paper, we study the geometric and dynamical properties of maximal representations of surface groups into Hermitian Lie groups of rank 2. Combining tools from Higgs bundle theory, the theory of Anosov representations, and pseudo-Riemannian geometry, we obtain various results of interest. We prove that these representations are holonomies of certain geometric structures, recovering results of Guichard and Wienhard. We also prove that their length spectrum is uniformly bigger than that of a suitably chosen Fuchsian representation, extending a previous work of the second author. Finally, we show that these representations preserve a unique minimal surface in the symmetric space, extending a theorem of Labourie for Hitchin representations in rank 2.

Rational Singularities, Quiver Moment Maps, and Representations of Surface Groups

Abstract We prove using jet schemes that the zero loci of the moment maps for the quivers with one vertex and at least two loops have rational singularities. This implies that the spaces of representations of the fundamental group of a compact Riemann surface of genus at least two have rational singularities. This has consequences for the numbers of irreducible representations of the special linear groups over the integers and over the $p$-adic integers.

$$\mathrm {SO}(p,q)$$-Higgs bundles and Higher Teichmüller components

Some connected components of a moduli space are mundane in the sense that they are distinguished only by obvious topological invariants or have no special characteristics. Others are more alluring and unusual either because they are not detected by primary invariants, or because they have special geometric significance, or both. In this paper we describe new examples of such `exotic' components in moduli spaces of SO(p,q)-Higgs bundles on closed Riemann surfaces or, equivalently, moduli spaces of surface group representations into the Lie group SO(p,q). Furthermore, we discuss how these exotic components are related to the notion of positive Anosov representations recently developed by Guichard and Wienhard. We also provide a complete count of the connected components of these moduli spaces (except for SO(2,q), with q> 3).

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).