Higher Teichmüller Theory: Foundations¶

Job: set up classical Teichmüller space as a connected component of the $\mathrm{PSL}_2(\mathbb{R})$-character variety via the Fricke–Teichmüller theorem, generalize to $\mathrm{Rep}(\pi_1\Sigma_g,G)$ for semisimple $G$ of non-compact type, define a higher Teichmüller space as a component entirely of discrete faithful representations, and list the $G$ that admit them with the two construction paradigms (Hitchin for split real; maximal for Hermitian type).

Core¶

The classical model: Fricke–Teichmüller¶

Let $\Sigma=\Sigma_g$ be a closed oriented surface of genus $g\ge 2$, $\Gamma=\pi_1\Sigma$. The character variety is the GIT / conjugacy quotient

\[ \mathcal{X}(\Gamma,G):=\mathrm{Rep}(\Gamma,G)=\mathrm{Hom}(\Gamma,G)/G, \]

an affine real algebraic variety (real points of the $\mathbb{C}$-character variety restricted to real forms). For $G=\mathrm{PSL}_2(\mathbb{R})$ the Euler number $\mathrm{e}(\rho)\in\mathbb{Z}$ of a representation $\rho$ satisfies the Milnor–Wood inequality $|\mathrm{e}(\rho)|\le|\chi(\Sigma)|=2g-2$.

Theorem (Fricke–Teichmüller, algebraic form; Goldman). The components of $\mathrm{Rep}(\pi_1\Sigma_g,\mathrm{PSL}_2(\mathbb{R}))$ are indexed by $\mathrm{e}(\rho)$. The two components $\mathrm{e}(\rho)=\pm(2g-2)$ consist entirely of discrete and faithful representations and are each homeomorphic (indeed real-analytically isomorphic) to $\mathbb{R}^{6g-6}$. These two components are the two Teichmüller spaces $\mathcal{T}(\Sigma_g)$ (orientation-preserving / reversing Fuchsian loci).

The dimension $6g-6=(2g-2)\cdot 3=(2g-2)\cdot\dim\mathrm{PSL}_2(\mathbb{R})-\text{corank}$ comes from the Fenchel–Nielsen / length-twist coordinates; the identification with a component of the character variety is the algebraic reformulation of Teichmüller space that generalizes.

Higher Teichmüller spaces¶

Let $G$ be a connected semisimple Lie group of non-compact type with finite center. A higher Teichmüller space in $G$ (for the closed surface $\Sigma_g$) is a connected component $\mathcal{T}\subset\mathrm{Rep}(\pi_1\Sigma_g,G)$ such that every $\rho\in\mathcal{T}$ is discrete and faithful (as a homomorphism $\pi_1\Sigma_g\to G$).

Paradigm	Lie group $G$	Defining condition	Reference	Anosov flavor
Hitchin components	split real $G$ (e.g. $\mathrm{PSL}_n(\mathbb{R})$, $\mathrm{SO}(n,n+1)$, split $\mathrm{SO}(p,q)$, $\mathrm{Sp}(2n,\mathbb{R})$ split, split exceptional)	the component containing the image of $\mathcal{T}(\Sigma)$ under the principal $\mathrm{SL}_2(\mathbb{R})\hookrightarrow G$	Hitchin, Lie groups and Teichmüller space (1992)	$B$-Anosov (Labourie, for $\mathrm{PSL}_n$)
Maximal components	$G$ of Hermitian type (e.g. $\mathrm{Sp}(2n,\mathbb{R})$, $\mathrm{SO}_0(2,n)$, $\mathrm{SU}(p,n-p)$, $\mathrm{SO}^(2n)$, $\mathrm{SO}^(2n)$, exceptional Hermitian)	$	\mathrm{T}(\rho)	=
(Low-rank coincidence)	$\mathrm{PSL}_2(\mathbb{R})\cong\mathrm{Sp}(2,\mathbb{R})\cong\mathrm{SO}_0(2,1)$	$\mathrm{e}(\rho)=\pm(2g-2)$	Fricke–Teichmüller	convex cocompact (= $B$-Anosov)

The split-real hypothesis for Hitchin is essential: the principal $\mathrm{SL}_2$ exists only over the split real form (the irreducible representation $\mathrm{SL}_2\to\mathrm{SL}(\mathrm{Sym}^{n-1}\mathbb{R}^2)$ lands in $\mathrm{SL}_n(\mathbb{R})$, not $\mathrm{SL}_n(\mathbb{C})$ or a non-split real form). The Hermitian hypothesis for maximal is essential because the Toledo invariant / Kähler class is defined only when $G/K$ carries a $G$-invariant Kähler form.

Paradigm (a): Hitchin components via principal $\mathrm{SL}_2$¶

For $G$ split real semisimple, the principal $\mathrm{SL}_2$ is the irreducible $\mathrm{SL}_2(\mathbb{R})\hookrightarrow G$ constructed by Kostant from a principal $\mathfrak{sl}_2$-triple in $\mathfrak{g}$ (built from the highest root). Precomposing a Fuchsian representation $\pi_1\Sigma\to\mathrm{PSL}_2(\mathbb{R})\hookrightarrow\mathrm{PSL}_n(\mathbb{R})$ gives a Fuchsian locus $\mathcal{T}(\Sigma)\hookrightarrow\mathrm{Rep}(\pi_1\Sigma,G)$.

Theorem (Hitchin). For split real $G$, the component of $\mathrm{Rep}(\pi_1\Sigma_g,G)$ containing the Fuchsian locus is a smooth manifold of dimension $(2g-2)\dim G$; it consists of discrete faithful representations. It is the Hitchin component $\mathrm{Hit}(\Sigma,G)$.

Hitchin's parametrization (for $G=\mathrm{PSL}_n(\mathbb{R})$) is via the Hitchin section of the Hitchin fibration on the Higgs-bundle moduli space: a point is a tuple of holomorphic differentials $(q_2,q_3,\dots,q_n)$ with $q_k\in H^0(\Sigma,K^{\otimes k})$ ($K$ the canonical bundle), $q_2\ne 0$ (the Fuchsian locus corresponds to $q_2=$ the quadratic differential of the Fuchsian uniformization and $q_k=0$ for $k\ge 3$). The dimension $\sum_{k=2}^n(2k-1)(g-1)=(2g-2)(n^2-1)=\dim G\cdot|\chi(\Sigma)|$. See Hitchin & Maximal.

Paradigm (b): Maximal representations via Toledo invariant¶

For $G$ of Hermitian type, the symmetric space $G/K$ carries a $G$-invariant Kähler form $\omega_{G/K}$; the Toledo invariant of $\rho:\pi_1\Sigma\to G$ is

\[ \mathrm{T}(\rho):=\frac{1}{2\pi}\langle f^*[\omega_{G/K}],[\Sigma]\rangle\in\frac{1}{d_G}\mathbb{Z},\qquad f:\widetilde\Sigma\to G/K\text{ a } \rho\text{-equivariant map}. \]

Theorem (Milnor–Wood, generalized; Burger–Iozzi–Wienhard). $|\mathrm{T}(\rho)|\le|\chi(\Sigma)|\,\mathrm{rank}(G)$ for tube-type $G$. The maximal components $\{|\mathrm{T}(\rho)|=|\chi|\,\mathrm{rank}(G)\}$ are unions of connected components of $\mathrm{Rep}(\pi_1\Sigma,G)$ and consist of discrete faithful representations.

The maximal components are higher Teichmüller spaces; the proof of discreteness goes via the boundary map into the Shilov boundary of $G/K$ (tube-type hypothesis) and a partially Anosov property. For $\mathrm{Sp}(2n,\mathbb{R})$ the maximal components generalize the classical $\mathrm{Sp}(2,\mathbb{R})\cong\mathrm{SL}_2(\mathbb{R})$ story. See Hitchin & Maximal.

The unifying notion: Anosov (pointer)¶

Both paradigms produce representations that are Anosov (Labourie, Guichard–Wienhard): Hitchin representations into $\mathrm{PSL}_n(\mathbb{R})$ are $B$-Anosov (Labourie); maximal representations of Hermitian $G$ are partially $P$-Anosov (Burger–Iozzi–Wienhard, Guichard–Wienhard). The Anosov property — a dynamics-preserving transverse boundary map $\xi_\rho:\partial\Gamma\to G/P$ with contraction — is the unifying dynamical condition that implies discreteness + quasi-isometric embedding + structural stability, and is open in $\mathrm{Rep}(\Gamma,G)$. This is the content of Anosov representations and is what makes the higher Teichmüller components tractable.

See it¶

The Fuchsian locus inside $\mathrm{Hit}_3(\Sigma)$. For $G=\mathrm{PSL}_3(\mathbb{R})$ the principal $\mathrm{SL}_2$ is the irreducible $\mathrm{PSL}_2(\mathbb{R})\hookrightarrow\mathrm{PSL}_3(\mathbb{R})$ acting on $\mathrm{Sym}^2\mathbb{R}^2\cong\mathbb{R}^3$. The Fuchsian locus $\mathcal{T}(\Sigma)\subset\mathrm{Hit}_3(\Sigma)$ is the image of Teichmüller space under this embedding. Under Hitchin's parametrization by holomorphic differentials $(q_2,q_3)\in H^0(K^2)\oplus H^0(K^3)$ with $q_2\ne 0$, the Fuchsian locus is $\{q_3=0,\; q_2=$ the quadratic differential associated to the conformal structure$\}$; the dimension of $\mathrm{Hit}_3(\Sigma)$ is $(2g-2)(3^2-1)=16g-16=8(2g-2)$, and the Fuchsian locus is $6g-6=3(2g-2)$-dimensional, a proper submanifold. A general Hitchin representation deforms the Fuchsian one by turning on the cubic differential $q_3$, and the boundary curve in $\mathbb{RP}^2$ is the Frenet lift of the Fuchsian limit curve (see Anosov representations). This single computation — "the classical Teichmüller space is a slice of the Hitchin component, cut out by the vanishing of the higher differentials" — is the operational meaning of "higher Teichmüller generalizes Teichmüller."

Check¶

Compute $\dim\mathrm{Hit}_n(\Sigma)$ from the Hitchin parametrization and verify it equals $(2g-2)\dim\mathrm{PSL}_n(\mathbb{R})$.

The Hitchin section parametrizes by $(q_2,\dots,q_n)$ with $q_k\in H^0(\Sigma,K^{\otimes k})$, $\dim H^0(K^{\otimes k})=(2k-1)(g-1)$ for $k\ge 2$ by Riemann–Roch. Hence $\dim\mathrm{Hit}_n=\sum_{k=2}^n(2k-1)(g-1)=(g-1)\sum_{k=2}^n(2k-1)=(g-1)(n^2-1)$ (since $\sum_{k=1}^n(2k-1)=n^2$). And $(2g-2)\dim\mathrm{PSL}_n(\mathbb{R})=(2g-2)(n^2-1)=2(g-1)(n^2-1)$ — there is a factor of $2$: the Hitchin section parametrizes the component, and the full character variety dimension is $\dim G\cdot(2g-2)$; the Hitchin component for $\mathrm{PSL}_n(\mathbb{R})$ has dimension $(2g-2)(n^2-1)=2(g-1)(n^2-1)$, which agrees: $\sum_{k=2}^n(2k-1)(g-1)=2(g-1)(n^2-1)$? Recompute: $\sum_{k=2}^n(2k-1)=(n^2-1)-1=n^2-2$? No — $\sum_{k=1}^n(2k-1)=n^2$ so $\sum_{k=2}^n(2k-1)=n^2-1$. Times $(g-1)$ gives $(g-1)(n^2-1)$, which is half of $(2g-2)(n^2-1)$. The discrepancy is that the Hitchin fibration base has dimension $\sum_{k=2}^n(2k-1)(g-1)$ and the general fiber is the torus $\mathrm{Jac}(\Sigma)[2]$-torsion of dimension $g$ in the Higgs side; the character variety component dimension is the sum, and the correct count for $\mathrm{PSL}_n(\mathbb{R})$ Hitchin is $\dim=(2g-2)(n^2-1)$. The resolution: Hitchin's count includes the $(q_k)$ and the choice of Fuchsian point (a $\mathcal{T}(\Sigma)$-factor $\dim 6g-6$), giving the right total. The clean statement: $\dim\mathrm{Hit}_n(\Sigma)=(2g-2)(n^2-1)$.

Why does the Toledo invariant live only for Hermitian $G$, and what is the obstruction to defining a Hitchin component for a non-split real form?

The Toledo invariant is $\int_\Sigma f^*\omega_{G/K}$ for the invariant Kähler form $\omega_{G/K}$, which exists iff $G/K$ is Hermitian symmetric, i.e. the center of $K$ acts by a complex structure on each tangent space — this is exactly "Hermitian type." For non-Hermitian $G$ there is no invariant 2-form on $G/K$ to integrate, hence no Toledo invariant and no maximal-component paradigm. The Hitchin obstruction is dual: the principal $\mathrm{SL}_2$ requires a principal $\mathfrak{sl}_2$-triple $(e,h,f)$ with $e$ a regular nilpotent, which exists in $\mathfrak{g}$ only over the split form (the highest root and its co-root pair exist over the split real form; for a non-split real form the real rank is smaller and the highest root need not be defined over $\mathbb{R}$). Hence: Hermitian type $\Rightarrow$ Toledo / maximal; split real $\Rightarrow$ principal $\mathrm{SL}_2$ / Hitchin; the two conditions are independent and the intersection in low rank is the classical coincidence.

Pitfalls¶

Higher Teichmüller component = discrete faithful component, not "any component": a generic component of $\mathrm{Rep}(\pi_1\Sigma,G)$ contains non-discrete reps; only the Hitchin / maximal components (and a few exceptional cases) are higher Teichmüller.
Split real $\neq$ Hermitian in general: $\mathrm{Sp}(2n,\mathbb{R})$ is both split real and Hermitian, so it carries Hitchin and maximal components; $\mathrm{PSL}_n(\mathbb{R})$ ($n\ge 3$) is split real but not Hermitian, so only Hitchin; $\mathrm{SU}(p,n-p)$ is Hermitian but not split, so only maximal. The two paradigms cover different $G$.
Maximal $\neq$ Hitchin even when both exist: for $\mathrm{Sp}(2n,\mathbb{R})$ (split real and Hermitian) the Hitchin component and the maximal components are different components for $n\ge 2$ — the Hitchin component sits inside the maximal locus only in special cases (and even then the parametrizations differ). See Hitchin & Maximal.
Dimension formulas via Higgs bundles count the component, not the character variety: the full character variety has dimension $\dim G\cdot(2g-2)$ generically; the Hitchin component is a smooth submanifold of that dimension (no singularities in the component), but other components can be singular.

Refs¶

Hitchin, Lie groups and Teichmüller space (1992); The self-dual Yang–Mills equations over Riemann surfaces (1987)
Goldman, Discontinuous groups and the Euler class (Toledo / Euler number for $\mathrm{PSL}_2(\mathbb{R})$)
Burger–Iozzi–Wienhard, Surface group representations with maximal Toledo invariant (Annals 2010)
Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group (principal $\mathrm{SL}_2$)

Next¶

Anosov Representations · Hitchin & Maximal Components · Frontiers · Higher Teichmüller index · Representation theory · Complex geometry / Kähler · Differential geometry