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# Projective representations of the Thurston geometries

homogeneous geometry is a pair (X, G) where X is a simply-connected Riemannian manifold and G is a maximal group of isometries of X with compact point stabilizers.

There are eight 3-dimensional homogeneous geometries, often called the Thurston geometries:

• Euclidean 3-space $\mathbb{E}^3$, hyperbolic 3-space $\mathbb{H}^3$, and the 3-sphere $S^3$;
• the product geometries $S^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$; and
• the more exotic fibered geometries $\widetilde{\mathrm{SL}_2\mathbb{R}}$Nil, and Sol.

The goal of this post is to show how all of these geometries can be modeled as projective geometries: more precisely, for any of these eight geometries (X, G), we can find some open domain $\Omega \subset \mathbb{RP}^3$ such that $\Omega \cong X$ (possibly up to index 2), and a subgroup $\Gamma \leq \mathrm{PSL}(4,\mathbb{R})= \mathrm{Aut}(\mathbb{RP}^3)$ such that $\Gamma \cong G$ (again, possibly up to index 2), and $\Gamma$ is the maximal subgroup of $\mathrm{PSL}(4,\mathbb{R})$ which preserves $\Omega$.

In other words, up to finite index and coverings, each of these geometries embeds (or, if we eliminate the preceding qualification, locally embeds) into 3-dimensional projective geometry.

I will follow the approach of Emil Molnár’s 1997 paper, which, for some reason, I seem to have great difficulty reading. Hopefully the process of writing this post / reading the article to reprocess the details for this purpose will help.

To describe the goal of this post somewhat more precisely: I wish to exhibit examples of such projective representations, or rather sketch semi-impressionistic outlines for how to obtain them. Readers in need of more (concrete, computational) details should consult Molnár paper; this post will hopefully provide a useful broad guide to the paper, but is not meant to be a substitute.

### The general setup: projective space and symmetric bilinear forms

We recall the definition of $\mathbb{RP}^3$ as the projectivization of a 4-dimensional vector space $V_4$, i.e. $\mathbb{RP}^3 := V_4 / \sim$ where we identify vectors in $V_4$ which are related by (real) scalar multiplication.

We also recall that $\mathrm{PSL}(4,\mathbb{R}) = \mathrm{PGL}(4,\mathbb{R})$ is in fact the full group of collineations in this case, since the Galois group $\mathrm{Gal}(\mathbb{R} / \mathbb{Q})$ is trivial.

The general game now is to consider symmetric bilinear forms $\langle \cdot , \cdot \rangle$ on $V_4$, of varying signatures, and possibly preserving certain fiberings in the case of the product / fibered geometries; then we take $\Omega$ to be some sort of invariant set w.r.t. $\langle \cdot , \cdot \rangle$, and $\Gamma$ to be the orthogonal (sub)group w.r.t. $\langle \cdot , \cdot \rangle$ (in $\mathrm{PGL}(4,\mathbb{R})$.)

We note that Molnár’s paper sets up a bunch of additional notation, as well as the notion of a polarity (which is equivalent to the bilinear form, via the dual vector space.) It appears that this is used to perform more explicit computations and more generally characterize projective representations of the various Thurston geometries, but are not strictly necessary for our purposes.

### The non-fibered geometries

Spherical geometry can be represented by taking a (positive- or negative-) definite bilinear form. Then the isotropy group $\Gamma$ to be [a projectivization of] the orthogonal group $\mathrm{O}(4)$, and we may take $\Omega$ to be all of $\mathbb{RP}^3$, or even better all of its better cover $S^3$.

Hyperbolic geometry can be represented by considering the hyperboloid model: take a Lorentz form, i.e. a form $\langle \cdot, \cdot \rangle$ with signature (-+++), let $\Omega$ be the projectivization of $\{p \in V_4 : \langle p, p \rangle < 0\}$; then the isotropy group is isomorphic to (the appropriate finite-index subgroup of) $\mathrm{SO}(1,3)$.

(3-dimensional) Euclidean geometry may be represented by considering any affine chart, and the corresponding affine group sitting inside $\mathrm{O}(4)$; the bilinear form in this case is degenerate, with signature (0+++).

### The need for additional constraints

For $\widetilde{\mathrm{SL}_2\mathbb{R}}$, we start with a bilinear form of signature (–++), but observe (after some work) that the point stabilizers under the orthogonal group are not compact.

We would now like to impose additional structure on our bilinear form, coming from the structure of our homogeneous geometry $\widetilde{\mathrm{SL}_2\mathbb{R}}$, which will whittle down the orthogonal group to a more manageable size.

This additional structure comes in the form of a non-trivial line bundle structure on the unit sphere $\Omega$ for our bilinear form, with base space a hyperboloid $\mathcal{H}$ and fibres given the skew lines given by the S-orbits of $\Omega$, where $S = \left\{ \left( \begin{array}{cc} R_\theta \\ & R_{-\theta} \end{array} \right) : \theta \in \mathbb{R} \right\} \cong \mathrm{SO}(2)$ ($R_\theta$ being the rotation matrix $\left( \begin{array}{cc} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{array} \right) \in \mathrm{SO}(2)$) is what Molnár calls the screw collineation group.

We observe that $\tilde{S} \cong \mathbb{R}$ acts freely on the base space $\mathcal{H}$, and that we may describe the unit sphere $\Omega$ as the universal cover of the unit tangent bundle $T^1\mathcal{H}$.

The collineation group $\Gamma$ we seek is then given by elements of our orthogonal group which also preserve our line bundle; Molnár has a very concrete description of this group, in suitable (standard) bases, as

$\left\{ \left( \begin{array}{cc} \cosh r R_\psi & \sinh r R_\alpha \\ \sinh r R^*_{\alpha-\omega} & \cosh r R^*_{\psi+\omega} \end{array} \right), \left( \begin{array}{cc} \cosh r R^*_\psi & \sinh r R^*_\alpha \\ \sinh r R_{\alpha-\omega} & \cosh r R_{\psi+\omega} \end{array} \right) : \psi, \omega, \alpha \in \mathbb{R}, r \geq 0 \right\}$

where $R_\theta$ is the rotation matrix $\left( \begin{array}{cc} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{array} \right)$ and $R^*_\theta$ is the twisted rotation matrix $\left( \begin{array}{cc} \cos\theta & \sin\theta \\ \sin\theta & -\cos\theta \end{array} \right)$

### The product geometries

We adapt this approach, in an intuitively rather more straightforward manner, for the product geometries $S^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$.

For $S^2 \times \mathbb{R}$ we again consider a bilinear form with signature (0+++), but now take as $\Omega$ no longer an entire affine chart on which the form is non-degenerate, but rather a punctured affine chart $\mathbb{A}^3 \setminus \{O\}$, where O is a fixed origin, and also equip the resulting punctured affine space with a line bundle—or really, a product structure—, with base space homeomorphic to the 2-sphere, and fibres given by open half-rays pointing outwards from O.

This gives us a distinguished family of 2-spheres (those with centre O); the group $\Gamma$ is then, as before, the subgroup of the orthogonal group preserving this line bundle structure. It is generated by the $\mathrm{SO}(2)$-subgroup of isometries of the unit 2-sphere, dilatations in the $\mathrm{R}$ direction, and inversions in our distinguished family of 2-spheres.

Analogously, for $\mathbb{H}^2 \times \mathbb{R}$, we consider a bilinear form with signature (0-++), and let $\Omega$ be the subspace (in the topological, not vector space sense) of a maximal affine chart on which the form is non-degenerate given by a punctured hyperboloid cone $\{p \in \mathbb{A}^3 : \langle p, p \rangle < 0\} \setminus \{O\}$ where again O is a fixed origin in the affine chart. Analogous to the previous case, equip $\Omega$ with a line bundle (product structure), with base space homeomorphic to the hyperboloid, and fibres given by open half-rays in pointing outwards from O.

We may then obtain $\Gamma$ as the subgroup of the orthogonal group preserving the product structure; it is generated, analogously to the previous case, by the $\mathrm{SO}(1,2)$-subgroup of isometries of the hyperbolic 2-space, dilatations in the $\mathrm{R}$ direction, and inversions in the distinguished family of hyperboloids coming from the line bundle structure.

### Sol and Nil

For Sol, we again start with a bilinear form with signature (0-++). Now (and for Nil) we do not have any (Cartesian) product structure; we take as $\Omega$ the entirety of an affine chart, corresponding to a maximal subspace on which our form is non-degenerate, and attempt to impose additional structure on this $\Omega$.

For Sol this additional structure comes in the form of a (trivial?) parallel plane fibering; the collineation group is then generated by a plane reflection, and a half-turn, and the elements $\left\{ \left( \begin{array}{cccc} 1 & a & b & c \\ 0 & e^c & 0 & 0 \\ 0 & 0 & e^{-c} & 0 \\ 0 & 0 & 0 & 1 \end{array} \right): a, b, c \in \mathbb{R} \right\}$.

For Nil we start with a bilinear form of signature (000+), and take as $\Omega$ an affine chart corresponding to a maximal affine chart on which our form is degenerate. There is an additional structure on this affine 3-space which takes the form of a line bundle. To describe this line bundle, we recall the description of Nil as a Heisenberg group; the base spaces of this fibration are then the horizontal levels $\left\{ \left( \begin{array}{ccc} 1 & a & c \\ & 1 & b \\ & & 1 \end{array} \right) : a, b \in \mathbb{R} \right\}$ of constant height c, and the fibres the “vertical” lines $\left\{ \left( \begin{array}{ccc} 1 & a & c \\ & 1 & b \\ & & 1 \end{array} \right) : c \in \mathbb{R} \right\}$.

$\Gamma$ is then given by (a suitable representation of) the Heisenberg group extended by pointwise stabilizers (isomorphic to a finite extension of $\mathrm{SO}(2)$); Molnár describes it explicitly as

$\left\{ \left( \begin{array}{cccc} 1 & x & y & z \\ & \cos\omega & \sin\omega & x \sin\omega \\ & \mp\sin\omega & \pm\cos\omega & \pm x\cos\omega \\ & & & \pm 1 \end{array}\right) : (x,y,z) \in \mathbb{R}^3 \setminus \{(0,0,0)\}, \omega \in \mathbb{R} \right\}$.

(in a suitably-chosen basis.)

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# Higher Teichmüller Theory

## An algebraic viewpoint: character varieties

I have been describing the genus g Teichmüller space Teich(g) as the space of essentially different hyperbolic metrics on the topological surface $\Sigma_g$ of genus g, or the space of conformal structures on $\Sigma_g$.

It is also possible to give it an altogether more algebraic description, and it is from this viewpoint that the generalisation to “higher” Teichmüller theory is perhaps most easily seen, or at least most superficially obvious.

Recall a point in Teichmüller space is given by a(n equivalence of) pair(s) [(S, h)] where S is a hyperbolic surface of genus g and $h: \Sigma_g \to S$ is an orientation-preserving homeomorphism, which is understood to be an isometry by fiat.

This is equivalent to a choice of isomorphism between the fundamental groups $\pi_1(\Sigma_g) \to \pi_1(S)$. Now $\pi_1(S)$ acts by isometries (deck transformations) on the universal cover of S, which is the hyperbolic plane; thus our choice of isomorphism between fundamental groups gives rise to a representation of the fundamental group $\pi_1(\Sigma_g)$ as isometries of the hyperbolic plane, or, in other words, since the isometry group of the hyperbolic plane is (isomorphic to) $\mathrm{PSL}(2,\mathbb{R})$, a homomorphism $\pi_1(\Sigma_g) \to \mathrm{PSL}(2,\mathbb{R})$. These representations are sometimes called holonomy representations.

Thus we can identify Teich(g) with (some subspace of) $\mathrm{Hom}(\pi_1(\Sigma_g), \mathrm{PSL}(2,\mathbb{R}))$, or rather of a quotient thereof, to take account of the corresponding quotient by homotopy—specifically, a quotient by the conjugation action of $\mathrm{PSL}(2,\mathbb{R})$.

This quotient $\mathrm{Hom}(\pi_1(\Sigma_g), \mathrm{PSL}(2,\mathbb{R})) / \mathrm{PSL}(2,\mathbb{R})$ is often called a representation variety or character variety (although apparently nobody has written down a proof that it is a variety in the algebro-geometric sense; conversely, nobody has a proof that it is not a variety either. It seems like the name came about since $\mathrm{Hom}(\pi_1(\Sigma_g), \mathrm{PSL}(n,\mathbb{C})) / \mathrm{PSL}(n,\mathbb{C})$ is [using considerable machinery to handle the quotient] a variety, and also $\mathrm{Hom}(\pi_1(\Sigma_g), \mathrm{PSL}(2,\mathbb{R}))$ is certainly an algebraic variety by rather more elementary arguments.)

Which part of our character variety is Teichmüller space identified with? We can show that the embedding $[(S,h)] \to h_*$ is open and closed, and hence Teichmüller space is (identified with) a component of the character variety. We also note that holonomy representations are discrete and faithful, and furthermore, using the Margulis lemma, that discreteness and faithfulness are both closed and open conditions, so that the discrete and faithful representations form a connected component of the character variety—this is the component that is identified with Teichmüller space.

We can now ask if anything interesting happens when we replace $\mathrm{PSL}_2\mathbb{R}$ with a different (semisimple) Lie group, possibly of higher rank—this is what the “higher” in “higher Teichmüller theory” most directly refers to. What is the structure of the corresponding character variety? Can we describe any of its connected component in terms of geometric, topological or dynamical properties of interest to us?

### Example: Hitchin representations

A Hitchin component is the connected component of $\mathrm{Hom}(\pi_1(\Sigma_g), \mathrm{PSL}(n,\mathbb{R})) / \mathrm{PSL}(n,\mathbb{R})$ which contains the image of Teichmüller space in $\mathrm{Hom}(\pi_(\Sigma_g), \mathrm{PSL}(2,\mathbb{R}))$ under the irreducible representation $\mathrm{SL}(2,\mathbb{R}) \to \mathrm{SL}(n,\mathbb{R})$. Elements of the Hitchin component are known as Hitchin representations.

The theory of Hitchin components shares many properties with Teichmüller theory: Hitchin representations are discrete, faithful, and quasi-isometric embeddings; one can prove collar lemmas; there are various coordinate systems on them which generalize coordinates on Teichmüller space.

[[ future addition: Higgs bundles and “deep connections” with algebraic geometry, which I do not understand but would like to [at least a little]. ]]

### Why surface (or 3-manifold) group representations into semisimple Lie groups?

From MathOverflow: “… the universal cover together with the deck group action contain a lot of information about the manifold, and the representations of the group provide one way to extract it … The space of representations into $latex \mathrm{SL}(n,\mathbb{C})$ is naturally an algebraic variety equipped with an additional rich structure which can conceivably be used to produce invariants of the original manifold.”

## A geometric viewpoint: (G,X)-structures

We could also ask if (components in) our new character varieties $\mathrm{Hom}(\pi_1(\Sigma), G) / G$ parametrize geometric structures of any sort. Here we are implicitly (or perhaps to some extent explicitly) using a point of view first expressed in Klein’s Erlangen program and nowadays formulated using the notion of (G, X)-structures, in which a “geometry” is characterized primarily by its symmetries—or, more precisely, described by a connected, simply-connected manifold X together with a Lie group G of diffeomorphisms acting transitively on X with compact point stabilizers.

Thus for instance Euclidean geometry is described by $(\mathrm{SL}_n \mathbb{R} \ltimes \mathbb{R}^n, \mathbb{R}^n)$, or hyperbolic geometry by $(\mathrm{SO}(n,1), \mathbb{H}^n))$.

Thus if we can identify our Lie group G in the character variety $\mathrm{Hom}(\pi_1(\Sigma), G) / G$ as acting transitively with compact point stabilizers on some connected, simply-connected manifold X, we will have a description of (at least certain components of) the character variety as parametrizing (certain) (G,X)-structures on the surface (or 3-manifold, or n-manifold for higher n, though what is known as n increases diminishes very rapidly.)

For instance: when G was $\mathrm{PSL}_2\mathbb{R} \cong \mathrm{Isom}^+(\mathbb{H}^2)$, we could take $X = \mathbb{H}^2$, and the corresponding character variety—or rather the component thereof which consisted of discrete, faithful representations—, which, recall, is exactly Teichmüller space, then parametrizes hyperbolic structures on the surface $\Sigma$. Aha.

### Example: the third Hitchin component and convex real projective structures

$\mathrm{SL}_3 \mathbb{R}$ (or rather the central quotient $\mathrm{PSL}_3 \mathbb{R}$) is the automorphism (isometry) group of real projective space $\mathbb{RP}^2$, and indeed Choi and Goldman proved that the n = 3 Hitchin component parametrizes convex real projective structures on a surface.

Danny Calegari exposits at more length on this moduli space on his blog.

In general, though, it is not so easy to obtain descriptions of higher representation varieties—even of the Hitchin components with $n \geq 4$—in terms of intuitively-comprehensible (G,X)-structures; in that respect it is an open question to obtain descriptions of such a geometric flavour.

### (G,X)-structures and flat bundles

A (G,X)-bundle on a manifold M is a space E together with a (projection) map $E \to M$ whose fibers are homeomorphic to X, and which admits local trivialisations with transition maps in G. For example: (G,X)-bundles where X is a vector space $k^n$ and $G = mathrm{GL}(n,k)$ are vector bundles.

(G,X)-bundles are, from one perspective, yet another way of globally encoding a collection of locally X-like structures patched up by bits of G, and indeed we can systematically go between them and (G,X)-structures by taking into account two additional pieces of information:

1. flat connection, which may be visualized as a “horizontal” foliation transverse to the fibers, which are preserved by the transition / gluing maps on M, and
2. a section, i.e. a left inverse to the projection map, transverse to the fibers.

Given a (G,X)-structure on M, we have a flat (G,X)-bundle on M with fibers isomorphic to X and local trivialisations described by the (G,X)-structure charts $X \times U_i$, with flat connection described by the horizontal foliation on $X \times X$, together with a section—the diagonal section of $X \times X$—transverse to the foliation.

Conversely, given a flat (G,X)-bundle on M (a bundle equipped with such a flat connection is known as a flat bundle) together with a section of the bundle transverse to the fibers, we can effectively reverse the above process to obtain a (G,X)-structure on M: intuitively speaking, the flat connection helps us determine where M is inside the total space of the bundle, and the section specifies which bit of X locally models each region of M.

### Example: maximal representations

A rather different way of picking out a component of interest starts with Milnor’s observation, subsequently extended by Wood to the Milnor-Wood inequality, that the Euler number of any flat plane bundle over a hyperbolic surface $\Sigma$ is at most $-\chi(\Sigma)$ in absolute value. Goldman, in his doctoral thesis, proved that the representations $\pi_1(\Sigma) \to \mathrm{PSL}_2\mathbb{R}$ whose associated flat $(\mathrm{PSL}_2\mathbb{R}, \mathbb{H}^2)$-bundle has maximal Euler number $-\chi(\Sigma)$ are precisely those which are holonomy representations of hyperbolic structures.

In other words, the maximal level set of the Euler number invariant in this case is a component in the character variety (Teichmüller space) of geometric interest.

Motivated by this, we may consider other representation invariants, often similarly constructed using cohomology, and define maximal representations as representations in the maximal level set of these invariants, where bounds for these invariants analogous to the Milnor-Wood inequality exist and where the level sets are well-behaved.

[[ future addition: some actual examples ]]

These maximal representations of surface groups have been shown to have some good geometric and dynamical properties: for instance, they are discrete and faithful; they are quasi-isometric embeddings when they are representations of closed surface groups; they are Anosov (see below.)

## Another algebraic viewpoint: lattices

Fundamental groups of closed or indeed finitely-punctured surfaces (with genus at least 2) are lattices in $\mathrm{Isom}(\mathbb{H}^2) \cong \mathrm{PSL}_2 \mathbb{R}$; similarly fundamental groups of closed, or more generally finite-volume hyperbolic 3-manifolds are lattices in $\mathrm{Isom}(\mathbb{H}^3) \cong \mathrm{PSL}_2 \mathbb{C}$.

We can thus view our character varieties as spaces of representations of lattices into semisimple Lie groups, and ask what happens, in slightly greater generality, as we vary the Lie group/s from which we take our lattices and into which we our representations take them. Here there is a striking contrast between what happens in low rank / dimension, and what happens in higher dimension / rank.

Hyperbolic surfaces carry a great multiplicity of possible hyperbolic structures and deformations: a whole Teichmüller space’s worth of them. On the other hand, finite-volume hyperbolic 3-manifolds are extremely rigid: Mostow-Prasad rigidity states that any homotopy equivalence between finite-volume hyperbolic 3-manifolds is induced by an isometry. Even stronger rigidity results hold for higher-rank Lie groups: Margulis superrigidity states that, loosely speaking, any linear representation of an irreducible lattice in a higher-rank semisimple Lie group is induced by a representation of the ambient Lie group. In other words, the deformation spaces of such lattices are trivial.

The main moral of the story here seems to be that—to speak in imprecise terms for a moment—, should there still be any geometric structures that we can associate to the points in our character varieties of 3-manifold or higher representations, we should not expect them to be closed, finite-volume, or similarly tame.

## Dynamical developments: Anosov representations

The examples of higher Teichmüller spaces above, somewhat disparate though they may be, share certain common structures, first explicitly described by Labourie for Hitchin representations, and subsequently systematically developed for more general representations $\Gamma \to G$ of word-hyperbolic groups $\Gamma$ into semisimple Lie groups G in Guichard-Wienhard.

Roughly speaking, these structures may be described as pairs of transverse limit maps which pick out attracting and repelling spaces at each point on $\partial_\infty \Gamma$, the Gromov boundary of our word-hyperbolic group, in a continuous way. This in turn gives rise to a coarsely Anosov structure (not a technical term—the right technical term here being either “metric Anosov flow” or “Smale flow”, depending on whom you ask) on the Gromov geodesic flow, which provides the setting for dynamical arguments to further the geometry of our representation varieties.

### Cartan projections, dominated splittings, and domains of discontinuity

[[ future addition: subsequent work of G(GK)W and KLP / see also Bochi-Potrie-Sambarino. Include Wienhard’s description of positive, maximal, and “mixed” representations? ]]

This gives rise to an identification of representations in (the specified components of) our character varieties as holonomies of certain geometric structures. There appear to be considerable difficulties involved, however, in attempting to make the converse of such an identification effective, i.e. in determining whether a given representation is a holonomy of a geometric structure of the type in question.

### Pressure metrics

Given a space $\mathcal{M}$ of Anosov representations, we can associate to each representation $\rho \in \mathcal{M}$ a Hölder function $f_\rho$ given by a natural reparametrisation function for the Gromov geodesic flow associated to the representation.

There is now a natural dynamical invariant, the pressure, on the space of Hölder functions, which by some fairly heavy machinery from dynamics (the thermodynamic formalism) varies analytically; moreover, to each Hölder function $f_\rho$ above there is naturally associated a pressure-zero function $h(f_\rho) f_\rho$, where $h(f_\rho)$ is the topological entropy of the flow associated to $f_\rho$.

Now, again by the thermodynamic formalism, the Hessian of the map $\rho \mapsto h(f_\rho) f_\rho$ is a well-defined positive-semidefinite quadratic form on the representation variety $\mathcal{M}$. With considerably more work, Bridgeman-Canary-Labourie-Sambarino showed that it is in fact positive-definite, and hence defines a Riemannian metric on the representation variety in question.

When $\mathcal{M}$ is Teichmüller space, the pressure metric is equivalent to the Weil-Petersson metric, about which some things are known (although many are not.) In more general cases, the geometry of the pressure metric is a wide-open question and an area of active research. How can we describe geodesics in this metric? Is it complete, and if not what is its completion? Is the metric, in general, negatively-curved?

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