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Algorithms via geometry

Configuration spaces and their geometry

For a fixed natural number n and manifold X, the configuration space \mathrm{Conf}^n X is space of all tuples of n distinct points in X, i.e. we may think of it as the open manifold X^n - \Delta, where here $\latex \Delta$ denotes the extended diagonal consisting of all n-tuples where the values of at least two coordinates coincide.

Configuration spaces arise naturally in the consideration of certain problems in physics, and more recently and more concretely in robotics: here, for instance, the manifold X may be taken to be the space our n robots are moving around, and then \mathrm{Conf}^n X represents all the possible combinations (“configurations”) of positions which the robots can occupy at any point in time. A path in $latex \mathrm{Conf}^n X$ then represents a set of possible simultaneous trajectories, and so the study of the configuration space and its properties becomes useful to motion planning.

More broadly, we may consider more general configuration spaces which represent the possible positions, or more general states, of a given robotic (or other) system, and think of paths in these configuration spaces as ways for the system to move from one position / state to another.

Sometimes the configuration spaces can be CAT(0), or even cubulated (i.e., in this context, made homeomorphic or maybe even bi-Lipschitz to a CAT(0) cube complex), and that can be convenient for developing algorithms to solve corresponding problems in robotics.This is the case, for instance, for a model robotic arm operating in a tunnel.

A slightly different application in which non-positively curved configuration spaces have also made an appearance is to phylogenetic trees.

Comparisons in image spaces

Suppose we have a collection of images of comparable two-dimensional surfaces, say neural images of the exterior surface of the human brain, or shapes of various animal bones, and we wish to compare them in some way, or perhaps obtain some “best match” mapping between a given neural image and some standard structural template/s.

This sounds very much like the question asked and addressed by Teichmüller theory: what are the natural maps between two different geometries on the same topological surface? In the case of higher-genus surfaces, the answer that Teichmüller theory gives is “extermal quasiconformal maps”, and more generally, and indeed something which forms an important part of the backdrop for Teichmüller theory in the first place, uniformization tells us that any compact surface is conformally equivalent to a sphere, torus, or hyperbolic surface.

Hence conformal maps and their geometry become natural tools in this sort of shape analysis, and quasiconformal maps (and possibly Teichmüller theory) can come into the picture when either discrete approximations or higher-genus surfaces are involved. Notions of distance between shapes, based on some notion of deformation energy, or on quasiconformal constants, can be useful in making quantitative statements such as “this shape is closer to model shape A than to model shape B.”

Cryptography from geometric group theory

Non-commutative cryptography uses cryptographic primitives, methods and systems based on non-commutative algebraic structure, as opposed to most familiar cryptographic systems, which are based on the commutative algebra underlying number theory.

Many of the protocols are very similar or analogous to those in more familiar (“commutative”) cryptographic systems, the main difference being that the hard (or, in many cases, presumed-to-be-hard based on existing evidence—rigorous cryptanalysis is still in many cases an open problem) problems underlying the security of the proposed primitives and/or systems come from group theory, or the theory of other non-commutative algebraic structures.

For instance, here is a general protocol for encryption / decryption: let G be a group, and let and B be commuting subgroups—i.e. ab = ba for all a \in A, b \in B, or in other words A \subset N_G(B), B \subset N_G(A), and fix x \in G (to be made public.) Alice and Bob choose secret keys a and b from A and publish y = x^a, z = x^b as public keys.

To send an encrypted message m, Alice picks a random r \in B and sends (x^r, H(z^r) \oplus m), where is some hash function and \oplus denotes XOR.

To recover the plaintext, Bob computes z^r = (x^b)^r = x^{br} = (x^r)^b and then m = H(z^r) \oplus m \oplus H(Z^r).

In the commutative setting, where we interpret G as (some finite quotient of) a integer ring and x^r as exponentiation, the security of this protocol would depend on the difficulty of finding discrete logarithms.

In the non-commutative setting, in a somewhat egregious abuse of notation, we interpret x^r as conjugation, and then the security of the protocol would depend on the difficulty of the conjugacy search problem (i.e. given z, t \in G, find r \in G s.t. t = rzr^{-1}.)

The difficulty of the conjugacy search problem in a given group, as well as other desirable properties (from either a security or an implementation standpoint), such as efficiently solvable word problem, computable normal forms, and super-polynomial growth function, is something that is often most (or at least more) easily studied using geometric methods.

Hence some of the groups which have been suggested in the context of this application: braid groups, Grigorchuk’s group, Thompson’s group/s, Artin groups, free groups, etc.

Other (apparently, or provably) difficult problems arising in geometric group theory may also be used, e.g. subgroup isomorphism in RAAGs (although this may potentially be less tenable in light of Babai’s recent breakthrough in graph isomorphism) or subgroup distortion in hyperbolic groups.

Non-commutative cryptography is still in many ways a nascent field; there are few concrete manifestations—the Algebraic Eraser is a rare example of one—and its security is presumed but yet to be fully tested—as demonstrated by the ongoing debate over the security of the Algebraic Eraser. Perhaps partly due to the relative lack of real-world applications, and partly due to the novelty of the field, cryptanalysis and work to establish the security of proposed protocols has been relatively slow in coming, although such work does exist.

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Weil-Petersson geometry (II)

Reformulations

Random geodesics and geodesic currents

Given two different hyperbolic metrics m and m’ on a closed topological surface, i.e. two different points in Teichmüller space, Thurston defined a quantity A(m, m') which can be interpreted as the length of a “random geodesic” in one metric measured in the other metric. A more precise definition involves intersection forms and Liouville measures and the transverse measure L_m to the geodesic flow, terms (some of) which are to be defined presently.

By the (rather computational) work of Wolpert, this metric turns out to be equivalent to the Weil-Petersson metric.

Bonahon reinterpreted this, yet again, in the language of geodesic currents. A geodesic current associated to a closed hyperbolic surface S is a measure on the set G(\tilde{S}) of unoriented geodesics of the universal cover \tilde{S} which is invariant under the natural action of \pi_1(S).

The space of geodesics G(\tilde{S}) embeds into the space of currents \mathscr{C}(S) as atomic measures, and this can be extended to an embedding of the measured laminations \mathscr{ML}(S) into \mathscr{C}(S). Indeed the image of the geodesics under scalar multiplication, i.e. \mathbb{R} \cdot G(\tilde{S}), is dense in \mathscr{C}(S).

Moreover, we can extend the geometric intersection number on closed geodesics to a continuous symmetric bilinear form i: \mathscr{C}(S) \times \mathscr{C}(S) \to \mathbb{R}^+.

What makes \mathscr{C}(S) an interesting object to consider is that there is also a natural embedding of Teichmüller space \mathcal{T}(S) \curvearrowright \mathscr{C}(S), which works by taking any point into the Liouville measure induced by the associated hyperbolic metric. This embedding is as nice as one might want it to be; in particular, its image can be characterized by analytic equations.

We can now use the intersection form to define a (path) metric on Teichmüller space, or rather on its image in the space of currents, relate this to Thurston’s metric, and then appeal to Wolpert’s result to find that this metric is in fact equivalent to the Weil-Petersson metric.

The thermodynamic formalism and the pressure metric

Yet another reinterpretation of—i.e. yet another way of defining a metric on Teichmüller space which turns out to be equivalent to—the Weil-Petersson metric involves the thermodynamic formalism from dynamics.

The thermodynamic formalism, for our purposes, is a machine from symbolic dynamics which produces analytically varying numerical invariants associated to families of (sufficiently nice) dynamical systems. In our case the dynamical system in question will be the geodesic flow on the unit tangent bundle of our closed hyperbolic surface, which may be represented as a finite-type shift using the Bowen-Series coding; we obtain our family of systems by considering Hölder reparametrizations of this flow.

From this system we obtain a space of \alpha-Hölder functions, which we view as reparametrizations of our geodesic flow, and thence a function C^{1+\alpha}(S^1) \to \mathbb{R}_{\geq 0}  the pressure function, which is equal to the log of the spectral radius of the transfer operator. Pressure-zero functions correspond to ergodic, flow-invariant equilibrium measures.

We may verify that pressure varies analytically, and so the second derivative of the pressure is well-defined, and is equal to the variance for a pressure-zero Hölder function. We may further check that the corresponding pressure metric defined by \|g\|^2_{\mathbf{P}} := \frac{\partial^2}{\partial t^2} \big|_{t=0} \mathbf{P}(f+tg) on the space of pressure-zero functions is positive-definite.

Working through this machinery, McMullen (building on the work of Bridgeman and Taylor, who extended the Weil-Petersson metric to quasifuchsian space) showed that the pressure metric may be expressed in terms of the Hessian of the Hausdorff dimension of the limit set, or of the pushforward measure on the boundary circle.

With more work, this Hausdroff dimension can also be related to the Weil-Petersson metric, and this can be used to show that the pressure metric on Teichmüller space is equivalent to the Weil-Petersson metric.

This pressure metric was subsequently extendedby Bridgeman, Canary, Labourie and Sambarino to more general higher Teichmüller spaces / spaces of Anosov representations. In that setting the technicalities are considerably more formidable—one has to produce a flow space to replace the geodesic flow on the unit tangent bundle, showing that the resulting pressure metric is well-defined and non-degenerate is much more work, etc.

Example here

[[ akan datang ]]

Applications

Wolpert’s expansions and estimates on curvature, together with the geometry of the boundary strata, can be used to produce an arithmetic Riemann-Roch formula for pointed stable curves, which yields e.g. the exact formula for the Selberg zeta function on the level-2 principal congruence subgroup \Gamma(2) < \mathrm{SL}_2\mathbb{R}. I don’t understand this at all well enough to comment further, but I’m going to speculate that the Weil-Petersson geometry plays the role which e.g. moduli space / stack arguments play in the Arakelov theory version.

A different (line of) application(s) appears in the groundbreaking work of Maryam Mirzakhani, who used a recursion formula for the Weil-Petersson volume of moduli space(s) to obtain asymptotic counts of simple closed geodesics on hyperbolic surfaces, establish relations to intersection theory on stable curves (including a new proof of the Witten-Kontsevich formula), applications to mathematical physics, and so on …

All of these results can also be used to study—going back to where it all began, in some sense—the properties of random Riemann surfaces of high genus.

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Weil-Petersson geometry (I)

We previously defined a Finsler metric on Teichmüller space, the Teichmüller metric; recall this was a L^\infty-type norm defined using extremal quasiconformal maps; it was complete, but not negatively-curved.

There is also a Riemannian metric on Teichmüller space, which is negatively-curved (although not in a terribly nice way) but not complete (although its metric completion has reasonably nice properties.) This is the Weil-Petersson metric, which we shall explore presently.

To define this metric, recall that we identified the tangent spaces to Teichmüller space as spaces of Beltrami differentials, or more precisely bounded (-1,1)-differentials (modulo infinitesimally-trivial ones.) Now we define the metric on the tangent space T_{[(S,h)]} \mathcal{T}_g by \langle \varphi, \psi \rangle_{WP} := \int \varphi\bar{\psi} \,ds, where ds denotes the hyperbolic metric on S.

Alternatively (and equivalently), we may define it first as a cometric, on the cotangent spaces T^*_{[(S,h)]} \mathcal{T}_g (which, recall, are identified with holomorphic quadratic differentials, i.e. (2,0)-differentials) by \langle \varphi, \psi \rangle_{WP} := \int \varphi\bar{\psi} \,(ds)^{-1}. Either of these expressions checks out formally, but why else on Earth would we consider an expression like that?

Perhaps the most natural way to arrive at such an expression (that I can think of, anyhow) is to start with the pairing T_{[(S,h)]} \mathcal{T}_g \times T^*_{[(S,h)]} \mathcal{T}_g \to \mathbb{C} given by (\varphi, \psi) \mapsto \int_S \varphi \psi: Given a Beltrami differential—which, recall, represents an infinitesimal deformation—we can feed it (pair it with) a holomorphic quadratic differential—which encodes infinitesimal changes under deformation—and average the resulting (infinitesimal) distortion across the surface. (This is also the [admittedly not altogether precise] sense in which the Weil-Petersson metric records a L^2-smoothed distortion, rather than the L^\infty-distortion that the Teichmüller metric records.)

But now any quadratic differential may be associated to a Beltrami differential by \psi \mapsto \psi (ds)^{-1}: again, this checks out formally in terms of degrees, and the correspondence doesn’t seem so unreasonable if we remember how both of these objects encode infinitesimal change in some way. Going through this correspondence and applying what comes out to our pairing results in the Weil-Petersson metric / cometric expressions above.

cf. the Petersson inner product on entire modular forms … apparently Weil first defined this metric taking inspiration from the Petersson inner product.

This post describes some salient geometric properties of this metric; a subsequent post will describe some of the reformulations / novel constructions of this metric, as well as applications of the Weil-Petersson geometry on Teichmüller space.

Kählerity and Fenchel-Nielsen coordinates

(full disclosure: this part somewhat shamelessly stolen off this blogpost of Carlos Matheus.)

The real part g_{WP} = \mathrm{Re} \langle \cdot, \cdot \rangle_{WP} induces a real inner product (also inducing the Weil-Petersson metric), while the imaginary part g_{WP} = \mathrm{Im} \langle \cdot, \cdot \rangle_{WP} induces an anti-symmetric bilinear form, i.e., a symplectic form\omega_{WP}.

By definition, if we let J denote the complex structure on Teich(S), we have g_{WP}(q_1, q_2) = \omega_{WP}(q_1, Jq_2). Moreover, as firstly discovered by Weil by means of a “simple-minded calculation” (“calcul idiot”) and later confirmed by Ahlfors and others (including McMullen, who produces an explicit Kähler potential), the Weil-Petersson symplectic form \omega_{WP} is closed, and so the Weil-Petersson metric is Kähler.

Using these properties, Wolpert (see also Section 7.8 in Hubbard’s book) showed that \omega_{WP} = \frac 12 \sum_{\alpha \in P} d\ell_\alpha \wedge d\tau_\alpha, where P is any pants decomposition on our surface, and \ell and \tau denote the corresponding length and twist parameters. In this sense, Fenchel-Nielsen coordinates are canonical (even if any particular manifestation of them involves an arbitrary choice of pants decomposition).

The proofs actually involve quite explicit considerations of twist deformations, and how the length parameters vary along these deformations. In particular, we have that the infinitesimal generator \partial / \partial\tau_\alpha of the twist about \alpha is the symplectic gradient for the Hamiltonian function \frac 12 \ell_\alpha, that is \frac 12 d\ell_\alpha = \omega_{WP}(-, \partial/\partial\tau_\alpha).

These considerations are also the starting point for Wolpert’s expansion formulas for the Weil-Petersson metric, which appear and are heavily useful below.

Geodesics

It is rather more difficult to describe Weil-Petersson geodesics than Teichmüller geodesics. (One reason, apparently, for a comment of Curt McMullen’s to the effect that the metric is “useless”.) Nevertheless we do know several large-scale properties:

The Weil-Petersson metric is uniquely geodesic.

Length functions, as well as their square roots, are convex along Weil-Petersson geodesics.

It is a result of Burns-Masur-Wilkinson that the Weil-Petersson geodesic flow is ergodic, so generically Weil-Petersson geodesics are equidistributed.

Incompleteness and metric completion

The Weil-Petersson metric is not complete: using Wolpert’s formula above to perform a first-order expansion of \omega_{WP} near a cusp, we may conclude that it is possible to degenerate a given curve \alpha to zero length within finite Weil-Petersson distance \sim\ell_\alpha^{1/2}

The metric completion of Teichmüller space with the Weil-Petersson metric is augmented Teichmüller space \overline{\mathcal{T}_g}—Teichmüller space with strata glued in consisting of points corresponding to noded surfaces, where some finite set \sigma of simple closed curves on our surface has degenerated to zero length; the set \sigma exactly determines the stratum \mathcal{T}_\sigma.

At a point X \in \overline{\mathcal{T}_g} near a stratum \mathcal{T}_\sigma, we have an adapted length basis, consisting of a set of curves (\sigma, \chi), where  \sigma is precisely the set of degenerated curves for the stratum (“short curves”, identifying the nearby stratum), and \chi is a collection of simple closed curves disjoint from those in \sigma, such that the tangent vectors \{d \ell_\alpha^{1/2}(X), J d\ell_\alpha^{1/2}(X), d\ell_\beta(X) \}_{\alpha \in \sigma, \beta \in \chi}, where J is the linear endomorphism on the T_X \mathcal{T}_g inducing the natural conformal structure on X, is a basis of T_X\mathcal{T}_g.

We can extend such a basis to a relative basis at X_\sigma \in \mathcal{T}_\sigma, now in the glued stratum, if the length parameters \{\ell_\beta\}_{\beta\in\chi} give a local system of coordinates for the stratum \mathcal{T}_\sigma near X_\sigma. In such a relative basis, we have Wolpert’s first-order expansion \langle \cdot, \cdot \rangle_{WP} \sim \sum_{\alpha \in \sigma} \left( (d\ell_\alpha^{1/2})^2 + (d\ell_\alpha^{1/2} \circ J)^2\right) + \sum_{\beta \in \chi} (d\ell_\beta)^2, and the implied constant is uniform in a neighborhood U \subset \overline{\mathcal{T}_g} of X_\sigma.

Curvature

Wolpert’s first-order expansion/s and “second-order Masur-type expansions” lead to the following estimates for any adapted length basis (\sigma, \chi) and any \alpha, \alpha' \in \sigma, \beta, \beta' \in \chi, with uniform constants on suitable Bers regions \Omega(\sigma,\chi,c) (regions where all the “short curves” in \sigma have length \geq \frac 1c, and all curves in \sigma \cup \chi have length \leq c.)

  • \langle d\ell_\alpha^{1/2}, d\ell_{\alpha'}^{1/2} \rangle = \frac 1{2\pi} \delta_{\alpha\alpha'} + O((\ell_\alpha\ell_{\alpha'})^{3/2});
  • \langle d\ell_\alpha^{1/2}, J d\ell_{\alpha'}^{1/2} \rangle =\langle J d\ell_\alpha^{1/2}, d\ell_\beta \rangle = 0;
  • \langle d\ell_\beta, d\ell_{\beta'} \rangle \sim 1, and this inner product extends continuously to the boundary stratum \mathcal{T}_\sigma;
  • \langle d\ell_\alpha^{1/2}, d\ell\beta \rangle = O(\ell_\alpha^{3/2}).

From these we obtain

  • d(X, \mathcal{T}_\sigma) = \sqrt{2\pi \sigma_\alpha \ell_\alpha} + O(\sum_{\alpha\in\sigma} \ell_\alpha^{5/2});
  • estimates of covariant derivatives;

and then the following estimates for sectional curvatures:

  • for any complex line \mathbb{R} d\ell_\alpha^{1/2} + \mathbb{R} J d\ell_\alpha^{1/2}, \langle R(d\ell_\alpha^{1/2}, Jd\ell_\alpha^{1/2} ) Jd\ell_\alpha^{1/2}, d\ell_\alpha^{1/2} \rangle = \frac 3{16\pi^2\ell_\alpha} + O(\ell_\alpha);
  • for any quadruple of vectors in the adapted length basis, not a curvature-preserving permutation of the previous quadruple, R(v_1, v_2)v_3, v_4 \rangle = O(1), and  each d\ell_\alpha^{1/2} or Jd\ell_\alpha^{1/2} in the quadruple introduces a multiplicative factor O(\ell_\alpha) in the estimate.

This yields (after non-trivial computation) that Teichmüller space with the Weil-Petersson metric is negatively-curved, although with sectional curvatures not bounded away from 0 or from -\infty. As a consequence the Weil-Petersson geometry is not coarsely (Gromov-)hyperbolic except for topologically simple surfaces (with 3g - 3 + n < 3)

Thus augmented Teichmüller space, being the metric completion of a uniquely geodesic negatively-curved space, has nonpositive curvature in the sense of the CAT(0) condition: triangles in this space are no fatter than they are in Euclidean space.

Curve complexes and pants graphs

The combinatorial structure of the strata in augmented Teichmüller space is determined by the set of simple closed curves which degenerate to zero length, and hence is described by the curve complex. (Recall the curve complex is the flag simplicial complex on the 1-skeleton with vertices corresponding to homotopy classes of simple closed curves on the surface, and edges between two vertices if the corresponding homotopy classes have disjoint representatives.)

Moreover, Brock and Margalit established that augmented Teichmüller space is quasi-isometric to the pants graph, which is a graph with vertices corresponding to pants decompositions (i.e. to maximal simplices in the curve complex), and edges between two vertices if the corresponding pants decompositions differ by replacing a curve \alpha with another curve \beta with minimal intersection number with \alpha; in other words, the Weil-Petersson metric, when considered between (maximally degenerate) strata, can be seen as coarsely encoding distances between the corresponding pants decompositions in the pants graph. The (highly non-unique) quasi-isometries in questions are given by sending a point X \in \mathcal{T}_g to a pants decomposition describing a Bers region X lies in, and conversely sending a pants decomposition to a point in Teichmüller space lying in the corresponding Bers region.

Symmetries

We may use the combinatorial structure described by the curve complex to describe the isometry group of the Weil-Petersson geometry as the extended mapping class group \mathrm{Mod}^\pm_g : Weil-Petersson isometries extend to the metric completion \overline{\mathcal{T}_g} and preserve the combinatorial structure of the strata, and hence preserve the combinatorial structure of curve complex. By a result of Ivanov, order-preserving bijections of the curve complex are induced by elements of the \mathrm{Mod}_g; hence there is a mapping class which yields our isometry on the maximally degenerate structure in \overline{\mathcal{T}_g}, and hence on the closed convex hull of these maximally degenerate structures, i.e. all of \overline{\mathcal{T}_g}.

The Nielsen-Thurston classification of elements of the \mathrm{Mod}_g, together with the general theory of isometries of CAT(0) spaces, then tell us a good deal about the geometry of these isometries: they either have fixed points in \overline{\mathcal{T}_g}, or else have positive translation length realized on a cloesd convex set isometric to a metric space product \mathbb{R} \times Y, on which our isometry acts as \mathrm{translation} \times \mathrm{id}_Y.

Alexandrov cones

There is a well-defined notion of angles between two geodesics emanating from a common initial point in a CAT(0) space, which allows to define tangent spaces in terms of sets of constant speed geodesics, modulo those at zero angle and having the same speed, with some natural topology on them.

In the interior of augmented Teichmüller space, the CAT(0) notion of angle coincides with the angles given by the Riemannian Weil-Petersson metric, and we get the usual tangent spaces; on the boundary stratum,  the CAT(0) Alexandrov angles yield Alexandrov tangent cones, which are isometric to Euclidean orthants (corresponding to the degenerate curvve “directions”) cross tangent spaces to strata.

These tangent-space-like structures have a number of applications here: e.g. they allow us to classify flat subspaces, yield a first variation formula for distance and non-refraction of geodesics (length-minimizing paths may change strata only at endpoints), allow us to construct combinatorial harmonic maps in certain cases, etc. (for slightly more detail, see Section 8 of Wolpert’s survey on Weil-Petersson metric geometry.)

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Bers simultaneous uniformisation

quasifuchsian group is a Kleinian group whose limit set is contained in an invariant Jordan curve. A quasifuchsian group of the first kind is one whose limit set is the whole of the invariant Jordan curve. The choice of the name “quasifuchsian” is justified by the following

Proposition: A Klenian group \Gamma is quasifuchsian iff it is quasiconformally conjugate (in \mathrm{PSL}_2\mathbb{C}) to a Fuchsian group.

Proof: The reverse direction is easy: boundary extensions of quasiconformal maps send limit sets of Fuchsian groups, which are round circles, to invariant Jordan curves.

The forward direction appears in a paper of Bers, and proceeds by noting that the domain of discontinuity \Omega(\Gamma) has two disjoint components, conformally mapping any one of them (with the boundary) to a round disc, and proving that we can extend this to a quasiconformal mapping of the other component.

Given a closed genus-g surface \Sigma_g, let QF(\Sigma_g) denote the space of quasifuchsian deformations of the fundamental group \pi_1(\Sigma_g), i.e. all conjugates of \pi_1(\Sigma_g) by quasiconformal maps.

Theorem (Bers). QF(\Sigma_g) \cong \mathcal{T}(\Sigma_g) \times \mathcal{T}(\overline{\Sigma_g}) (for g \geq 2.)

In other words, given any two Riemann surfaces of the same genus g \geq 2, there is some (hyperbolizable) 3-manifold which simultaneously uniformizes them, in the sense that its conformal boundary consists of two components, which are precisely the given Riemann surfaces.

(John Hubbard in his tome on Teichmüller theory compares the result to the works of Hieronymous Bosch: somehow unnatural and horrifying, but still a work of art. I don’t know if I agree, but it’s an interesting description anyhow … )

Proof: Let \Gamma := \pi_1(\Sigma_g), and define a map \Theta: QF(\Sigma_g) \to \mathcal{T}(\Sigma_g) \times \mathcal{T}(\overline{\Sigma_g}) by \phi\Gamma\phi^{-1} \mapsto \left( (\phi(S), \phi), (\bar\phi(S), \bar\phi) \right).

It is fairly straightforward to show that \Theta is surjective: given ((X,g), (Y,h)) \in \mathcal{T}(\Sigma_g) \times \mathcal{T}(\overline{\Sigma_g}), g \coprod h lifts to a quasiconformal map f: U \coprod L \to \mathbb{H}^2 \coprod \overline{\mathbb{H}^2}, where U and L denote the upper and lower half-planes resp. We may then check that the quasifuchsian group corresponding to f is sent to the point ((X,g), (Y,h)) \in \mathcal{T}(\Sigma_g) \times \mathcal{T}(\overline{\Sigma_g}) we started with.

To show \Theta is injective: suppose \Theta(\rho_1) = \Theta(\rho_2), so that there exists some conformal map j taking the conformal boundary of N_{\rho_1} to that of N_{\rho_2}, which lifts to a conformal map \tilde{j}: \Omega(\rho_1) \to \Omega(\rho_2) between the respective domains of discontinuity.

Now recall the limit sets \Lambda(\rho_1) and \Lambda(\rho_2) are both invariant Jordan curves, so we may canonically extend \tilde{j} over the limit sets by sending a point on one limit set (thought of as a direction at infinity) to the the corresponding point (direction at infinity) on the other limit set. This gives us a map \hat{j}: \hat{\mathbb{C}} \to \hat{\mathbb{C}}, which we may argue is quasiconformal.

Now an extension theorem of Douady-Earle allows us to naturally extend this quasiconformal map \hat{j}, or more specifically its quasisymmetric restriction to the limit sets, to a quasiconformal map of the interior disc (i.e. the quasiconformal copy of the hyperbolic plane) which it bounds. That such a map existed we already knew, but the naturality properties that come with the Douady-Earle extension, plus the fact that the quasiconformal \hat{j} is conformal on a subset of \hat{\mathbb{C}} of full measure and hence conformal, tells us that this induced extension is in fact conformal, i.e. in fact \rho_1(\Gamma) is conformally conjugate to \rho_2(\Gamma), as desired.

It is clear (or, at least, reasonable) that the map is continuous, as is its inverse, and we are done (a more rigorous proof of this last part would involve going through the construction of the map more carefully, and probably invoking the continuous dependence given by the measurable Riemann mapping theorem at some point.)

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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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Teichmüller geometry: a primer

The genus g Teichmüller space Teich(g) is the space of all marked hyperbolic metrics on a genus g surface \Sigma_g, modulo homotopy, or equivalently the space of all conformal structures, modulo biholomorphisms isotopic to the identity.

Recall it can be given a topology in a number of ways. Could we also give the space a geometry?

Infinitesimally speaking

In line with the general philosophy of deformation theory, we may attempt to get a handle on the local geometry of this space—if indeed it exists—by describing infinitesimal deformations of hyperbolic metrics or conformal structures.

How could we describe deformations? One way is to by using a pair of transverse measured foliations on our surface—roughly speaking, “nice” partitions of the surface (minus finitely many singular points) into 1-dimensional manifolds—which record the directions of maximal and minimal stretch of an deformation on the surface. These can be encoded, complex analytically, using holomorphic quadratic differentials.

Alternatively, and in some sense more directly, we may describe infinitesimal deformations using quasiconformal maps—continuous maps whose infinitesimal distortion is uniformly bounded, from both above and below—which, by the measurable Riemann mapping theorem (which despite its name is not directly related to the Riemann mapping theorem) may be encoded, complex analytically, using Beltrami differentials.

Now recall that tangent vectors are, in general, derivations, i.e. infinitesimal deformations, and cotangent vectors are what tangent vectors—which are canonically identifiable with cocotangent vectors—eat (for variety’s sake, they also eat Koko Krunch for breakfast.) With this in mind, we assert (without quite proving, although one could) that QD(S), the complex vector space of holomorphic quadratic differentials on S, is (or, rather, is isomorphic to, or in other words can be used to describe) the cotangent space to Teich(g) at [(S, h)], and BD(S), the complex vector space of Beltrami differentials modulo infinitesimally-trivial ones the tangent space.

Distances and geodesics

This characterization of the tangent and cotangent spaces allows us to define a metric, the Teichmüller metric, by d\left( [(S_1. h_1)], [(S_2, h_2)] \right) = \inf_f K_f, where the infimum is taken over all quasiconformal homeomorphisms f: S_1 \to S_2, and K_f is the dilatation of f.

By construction (and the properties of quasiconformal maps), this is symmetric and satisfies the triangle inequality; proving positivity requires Teichmüller’s existence and uniqueness theorems, which assert that there exists a unique (up to homotopy) minimal-dilatation quasiconformal map between any two points in Teichmüller space.

The Teichmüller metric is Finsler, not Riemannian: it is a L^\infty-type norm; more precisely, we may show (after some relatively straightforward, but non-trivial work) that the norm it induces on the tangent spaces is given by \mu \mapsto \sup \mathrm{Re} \int \mu q, where the supremum is taken over all holomorphic quadratic differentials q with \|q\| = 1.

(In particular, the norm is dual to a L^1-type norm on QD(S), is hence strictly convex, and in particular not something induced by a bilinear form—in other words, not Riemannian.)

With a metric in hand we can ask what the geodesics in this space look like, i.e. what are the “straight lines” (which are the geodesics in our familiar Euclidean space) in this space?

Well, Teichmüller geodesics (sometimes called Teichmüller lines) are given by fixing a holomorphic quadratic differential—or, from a geometric point of view, fixing a pair of transverse measured foliations on the surface—and then scaling that differential, i.e. continuously stretching and contracting along these foliations. What this does to the hyperbolic metric / conformal structure is fairly straightforward to visualize on a single coordinate patch, but globally, on the surface as a whole, the effect can be rather more complicated.

[One day I’ll have a visualization here. Maybe. Hopefully.]

This characterization of Teichmüller geodesics can also be used to show that the Teichmüller metric is (geodesically, and hence metrically by Hopf-Rinow) complete.

Curvature

The Teichmüller metric is not negatively-curved. Since we are not dealing with a Riemannian metric here, we need to specify what we mean here, and what we will sketch below is that the metric is not negatively-curved in the sense of Busemann: distinct geodesic rays originating from the same point do not diverge exponentially.

Masur constructed more-or-less explicit examples of distinct geodesic rays (in his thesis, which then became an Annals paper) which stay a bounded distance apart. To build these, he starts with a pair of Strebel differentials—holomorphic quadratic differentials whose horizontal foliations consist of closed leaves, which may be grouped into finitely many annuli, which together with a finite number of singular leaves form a partition of the surface—with the same annuli but different relative lengths between the annuli. (Ralph Strebel proved that such differentials always exist, with prescribed relative lengths, hence their name.)

Now do affine stretches along each annuli to obtain the geodesic rays, which are called Strebel rays. The construction of the Strebel differentials, and more precisely the structure of their horizontal foliations, ensures that corresponding points on the resulting rays are always bounded quasiconformal dilatation, i.e. bounded Teichmüller distance, apart.

Symmetries

We may also ask: what are the isometries of this space? Or, slightly less precisely but perhaps more descriptively, what symmetries does this space have?

To answer this, we define the mapping class group Mod(g) (sometimes called the Teichmüller  modular group) as the group of all orientation-preserving homeomorphisms \Sigma_g \to \Sigma_g, modulo those which are homotopic to the identity.

It is clear that any element of the mapping class group induces an isometry of Teichmüller space: pre-composing a Teichmüller map with any representative of mapping class still yields a Teichmüller map.

It is a result of Royden that the converse is also true: any isometry of Teichmüller space (modulo the hyperelliptic involution in the case of genus 2) comes from the mapping class group.

The key lemma involved, which Royden proves by mucking around with the space of holomorphic quadratic differentials and its analytic / algebraic underpinnings, is this: any complex linear isometry between the (co)tangent spaces induces a conformal map between the underlying points in Teichmüller space—which, recall, may be thought of as Riemann surfaces.

(In fact Royden’s lemma—really a theorem—characterizes these linear isometries very precisely: any such complex linear isometry \varphi is given by \varphi(\eta) = \alpha\eta \circ \psi, where \alpha \in \mathbb{C} has modulus 1 and \psi is a conformal map between the surfaces, although we will not need the full strength of it here.)

Now any isometry induces complex linear isometries between cotangent spaces and between tangent spaces, and hence conformal maps—in particular orientation-preserving self-homeomorphisms, which are well-defined mod homotopy, or in other words, mapping classes—between the underlying points.

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Deformation spaces and Teichmüller theory

So we have this thingummy, and we’re trying to find out more about it. What would you do? Shake it a little and see what happens, maybe? That’s one way you might describe the motivation for (or, at least, one motivation for) deformation theory. By figuring out how a geometric structure—a triangulation, say, or a foliation, or a smooth atlas, or a hyperbolic structure—can be perturbed, we learn more about the nature of the structure at hand.

Deformation spaces are not a precisely-defined concept; moduli spaces are slightly more precisely-defined, though not completely so, and in some sense a similar (though perhaps somewhat broader) idea.

One key example of a moduli space is the genus g Teichmüller space Teich(g): this is the space of all marked hyperbolic metrics on a genus g surface \Sigma_g (the genus g surface, when we regard it as the unique topological object in its homeomorphism class), modulo homotopy.

Here “marked” indicates that points in Teich(g) correspond not just to a choice of hyperbolic metric on \Sigma_g (described mathematically by e.g. an orientation-preserving homeomorphism from a “standard” / fixed surface with a metric which is declared to be an isometry), but also a choice of isomorphism to the fundamental group \pi_1(\Sigma_g)—not just hyperbolic clothing, but also instructions on how to wear it. The specification of a marking “rigidifies” and helps clarify the effect of automorphisms (i.e. non-trivial mapping classes) on our hyperbolic surface—something which is concern e.g. when we try to build a fine moduli space.

It is one of the great triumphs of 19th century mathematics that Teich(g) is also the space of all marked conformal structures on the genus g surface, modulo biholomorphisms homotopic to the identity. Any conformal structure yields a hyperbolic metric via uniformization; conversely, given a hyperbolic metric, we may produce a conformal structure using isothermal coordinates.

This dual identity allows Teichmüller theory—the theory which describes Teichmüller space and what it can do—to draw on tools from both hyperbolic geometry and complex analysis, and gives the theory much of its richness.

Setting out from the viewpoint of hyperbolic geometry, we may obtain, for instance, the Fenchel-Nielsen coordinates—which start by taking all-right hyperbolic hexagons, gluing these to get pairs of pants with specified cuff lengths, and then gluing pairs of pants with specified twists to get hyperbolic surfaces—and a proof of the Nielsen-Thurston classification, parts of which look at how lengths of curves on the hyperbolic surface change under the action of (various types of) mapping classes.

Setting out from the viewpoint of complex analysis, we obtain a systematic description of the tangent and cotangent spaces to Teichmüller space as spaces of various sorts of differential forms—Beltrami differentials and holomorphic quadratic differentials, respectively—, which in some sense encode representations of infinitesimal deformations of the metric structure—quasiconformal maps and transverse measured foliations, resp.

From there we may obtain results on optimal deformations between the various hyperbolic / conformal structures—this is the content of Teichmüller’s existence and uniqueness theorems, and the basis of Teichmüller geometry, i.e. the study of the geometry of Teich(g) as not just a collection of spaces, but a space in its own right.

I’ve written elsewhere, at some [mild] length, about moduli spaces in general and Teichmüller space in particular, including the motivation for these, so I should not continue on here. I did want, nevertheless, to emphasise the dual nature of the space and the tools involved—something I’ve come to appreciate more recently—, and to highlight the theory’s role as a base case and starting point for some of the subsequent areas I explore next: higher Teichmüller theory, and Weil-Petersson geometry.

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