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Translation surfaces and friends

Recall that we earlier defined translation surfaces as maximal atlases of coordinate charts into the Euclidean plane \mathbb{R}^2, with finitely many cone singularities and where the transition maps between coordinate charts are translations in \mathbb{R}^2.

Recall that we were led to this notion starting from polygonal billiards, but not every translation surface, generally defined, comes from a rational polygonal billiard.

Enter Riemann surfaces and moduli spaces

A translation surface may also be thought of as a Riemann surface, together with an associated holomorphic 1-form—a continuous choice of specified direction (“up”) at every point; such a pair of structures canonically defines a (singular) flat structure on the surface, with a distinguished vertical direction.

To wit: we take the holomorphic 1-form \omega to be (locally) our dz. Since \omega is holomorphic, its zeroes are isolated: a zero of degree d corresponds exactly to a conical singularity with cone angle 2\pi(d+1). Where \omega is not zero, it is represented (locally) by dz, whence we obtain a complex coordinate z, which in turn specifies (locally) a Euclidean structure. Since dz is globally well-defined on our translation surface, the resulting flat structure is also well-defined, away from the conical singularities.

Conversely, given a flat structure with conical singularities P_1, \dots, P_n, and a specified direction, consider a fundamental polygon P_1P_2 \cdots P_n embedded (anywhere, but with orientation dictated by the specified direction) in the complex plane. The fundamental polygon inherits a natural complex coordinate z. This does not descend to the translation surface, but since the identification maps are all of the form z \mapsto z + \zeta where \zeta \in \mathbb{C} is a constant (for each identification map), the holomorphic 1-form dz does, and we obtain a holomorphic 1-form \omega on our translation surface. \omega is zero exactly at the conical singularities, with cone angle being related to degree as above.

The Riemann surface perspective emphasizes more clearly the presence of a moduli space of translation surfaces (on for a fixed genus g), which we presently define as a space of pairs (X, \omega), where X denotes a Riemann surface structure, and \omega a choice of holomorphic 1-form, modulo some natural equivalence relation. The 1-form specifies a distinguished direction—so there is a well-defined notion of “north” on the surface. Two such pairs (X_1, \omega_1) and (X_2, \omega_2) are considered equivalent if there is a conformal map from X_1 to X_2 which takes (specified) zeroes of \omega_1 to (specified) zeroes of \omega_2.

The moduli space is divided into distinct strata \mathcal{H}(d_1, \dots, d_m), consisting of forms with zeroes of degree d_1, \dots, d_m with d_1 + \dots + d_m = 2g-2. This last identity follows from the formula for the sum of degrees of zeroes of a holomorphic 1-form on a Riemann surface of genus g, and can be interpreted as a Gauss-Bonnet formula for the singular flat metric.

There is a \mathrm{SL}_2\mathbb{R} action on this moduli space (or, really, on each stratum) which is most easily described back in the framework of flat geometry: given any pair (X,\omega), build the corresponding flat polygon (with distinguished vertical direction.) Elements of \mathrm{SL}_2\mathbb{R} act (linearly) on these flat polygons, thought of as being embedded in the Euclidean plane, with (say) some vertex at the origin. This action preserves parallelisms between edges; and hence is (induces) an action on the corresponding flat surfaces.

fig15

(figure from the Zorich survey—the \mathrm{SL}(2,\mathbb{R}) action, depicted on the left, changes the affine structure but not the translation structure; the “cut-and-paste” on the right changes the translation structure, but not the affine structure.)

We can study the dynamics of this action, and this turns out to be surprisingly (or perhaps not surprisingly—or maybe that is only with the benefit of giant-assisted hindsight) rich …

(There were already hints of this in the last post, when we talked about “changing the translation structure while preserving the affine structure” and how Masur used this idea to prove his theorem on counting closed geodesics.)

Counting closed geodesics and saddle connections

As previously noted in the slightly more restricted context of rational polygonal billiards, directional flow on a flat surface is uniquely ergodic in almost every direction.

This is most transparently seen in the case of a torus: geodesics with rational slopes are closed, while those with irrational slopes are equidistributed. Geodesics on flat surfaces of higher genera exhibit certain similiarities: the closed geodesics also appear in parallel families, although in higher genera these do not fill the whole surface, but only flat cylinders with conical singularities on the boundary.

Related to closed geodesic are saddle connections, which are geodesic segments joining two conical singularities (which may coincide), with no conical points in their interior. On a flat torus there are no conical singularities, and so any closed loop can be tightened to a closed geodesic; on a more general flat surface, this tightening process can produce either a closed geodesic, or—if at some point in the process we hit one of the singularities on the surface, which, as one might imagine, is the more generic case—a union of saddle connections.

Masur and Eskin have found quadratic asymptotics, as a function of length, for the number of [cylindrical families of] closed geodesics—this was discussed more in the previous post—and the number of saddle connections. The constants which appear in these asymptotics are called the Siegel-Veech constants. There are also (somewhat surprising) quadratic asymptotics for multiple cylindrical families in the same parallel direction.

These results are interesting in their own right—on a [rational] billiard table, for instance, generalized diagonals (trajectories joining two of the corners, possibly after reflections) unfold to saddle connections and periodic trajectories unfold to closed regular geodesics—but are also useful for at least two other reasons:

One, degeneration of “configurations” of parallel saddle connections or closed geodesics leads us into cusped regions in the boundary of (strata in the) moduli space, and so a description of such configurations gives us a description of the cusps of our strata. Local considerations involving short / degenerating saddle connections also lead us to relations between and structural results about the strata, which can be described more carefully / analytically in the language of Abelian differentials.

Two, configurations of saddle connections and closed geodesics are also useful as invariants of \mathrm{SL}(2,\mathbb{R}) orbits—something that we will refer back to below.

Volume of moduli space

We remark that this last counting problem can be related to computations of volumes of the moduli space, via the observation that the Siegel-Veech constants can be obtained as a limit of the form \lim_{\epsilon \to 0} \frac{1}{\pi \epsilon^2} \frac{\mathrm{Vol}(\epsilon\mbox{-neighborhood of cusp }\mathcal{C})}{\mathrm{Vol} \mathcal{H}_1^\circ(d_1, \dots, d_n)} where \mathcal{C} is a specified configuration of saddle connections or closed geodesics.

Athreya-Eskin-Zorich used this idea to obtain explicit formulas (conjectured by Kontsevich based on experimental evidence) for the volumes of strata in genus 0, by counting generalized diagonals and periodic trajectories on right-angled billiards. In general, of course, the relation between the two problems can be exploited in both directions: results on volumes of strata can also be used to obtain results on the counts of saddle connections / closed geodesics.

There are a number of alternative strategies for finding these volumes; the following is so far the most general:

The general idea is to simply use asymptotics for counts of integer lattice points, where the lattice is defined in terms of cohomological period coordinates. We can count the number of such points in a sphere or hyperboloid (which is the unit sphere for an indefinite quadratic form of suitable signature) to estimate its volume, and take a derivative to estimate the volume of the boundary hypersurface.

Integer lattice points may be thought of, geometrically, as flat surfaces tiled by square flat tori, and combinatorial geometric methods may be used to count these: in the simplest cases we can count directly; slightly more generally we can consider the graph with conical points as vertices and horizontal saddle connections as edges, leading to the notions of ribbon graphs and separatrix diagrams.

In the most general case we turn, following an idea of Eskin-Okounkov, to representation theory: suppose there are N squares; label them, and consider the permutation \pi on [N] which sends j to the square \pi(j) which we get to by starting at j and moving left, up, right, and down in turn. For the generic square j, \pi fixes j, but near the conical points it acts non-trivially, and indeed it is a product of m cycles of lengths (d_1+1), \dots, (d_m+1).

It then suffices to count the number of permutations of N with such a property … except there is a nontrivial correction needed to pick out only those permutations which correspond to connected square-tiled surfaces. Eskin-Okounkov-Pandharipande pushed through this strategy to obtain explicit quantitative results, with a strong arithmetic flavor. These results may be made explicit, although there is considerable computational work involved; by comparison, other approaches such as the work of Athreya-Eskin-Zorich referred to above can produce simpler formulas in special cases.

Applications

We have described above and previously how translation surfaces are related to (rational) polygonal billiards, and how the former provide a powerful framework for the study of the latter. Here (and in a subsequent post) we present a number of other applications:

Electron transport

S. P. Novikov suggested the following as a mathematical formulation of electron transport in metals: consider a periodic surface \widetilde{M^2} \subset \mathbb{R}^3; an affine plane in \mathbb{R}^3 intersects this in some union of closed and unbounded intervals. Question: how does an unbounded component propagate in \mathbb{R}^3 (as we move the affine plane in some continuous fashion?)

After we quotient by the period lattice (taken to be \mathbb{Z}^3), we are looking at plane sections of the quotient surface M^2 \subset \mathbb{T}^3. Our original intersection can be viewed as level curves of a linear function f(x,y,z) = ax + by + cz restricted to \widetilde{M^2}, but this does not push down to the quotient; instead, we consider the codimension-one foliation of M^2 defined by the closed 1-form df = a\,dx + b\,dy + c\,dz.

Our question can then be reformulated as follows: what do lifts of leaves of this foliation on M^2 \subset \mathbb{T}^3 look like upstairs, in \widetilde{M^2} \subset \mathbb{R}^3?

Closed 1-forms on surfaces can be straightened to geodesic foliations in appropriate flat metrics iff any cycle formed from a union of closed paths following a sequence of saddle connections is homologically non-trivial. The surfaces and 1-forms obtained from Novikov’s problem can be modified (decomposed and surgered) to satisfy this criterion, and after these reductions we are left exactly in the world of flat structures with closed 1-forms on them, i.e. translation surfaces.

Invisibility

In a subsequent post we describe the illumination problem; a related problem concerns invisibility—more precisely: whether a body with mirror surfaces can be “invisible” from some direction/s, because light rays travelling in these direction/s are reflected in such a way that they continue along trajectories in the same direction/s; or whether a body can be similarly made invisible in certain driections through the strategic placement of mirrors around it. It is a conjecture of Plakhov the set of directions that are invisible for any fixed body has measure zero. This conjecture is closely connected with the (similar) Ivrii conjecture on the measure of the set of periodic billiard trajectories in a bounded domain: if Ivrii’s conjecture is true then, most probably, true also is the conjecture on invisible light rays.

Flat (but not very), and real affine …

There are at least two ways in which the notion of translation surfaces can be generalized: one is to consider flat structures with non-trivial linear holonomy, which “forces a generic geodesic to come back and to intersect itself again and again in different directions.” The other to consider real affine structures, which are maximal collections of charts on a closed surface where all of the transition maps are of the form f(z) = az+b where a > 0 (and is in particular real) and b \in \mathbb{C}.

These remain rather less well-studied and more mysterious, or, in other words / from another perspective, present potentially rich sources of interesting open problems …

References

Zorich’s survey covers a broad range of ideas, and contains many further references

Hubert-Masur-Schmidt-Zorich have a (slightly outdated) list of open problems on translation surfaces , from a conference at Luminy.

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Polygonal billiards

Polygonal billiards are easily-described dynamical systems, which are defined by the trajectories of a single particle in a polygonal region P of the plane by requiring the particle move in straight lines at constant velocity in the interior of P, and reflect off the boundary according to the familiar (“optical”) laws of reflection: the angle of incidence should equal to the angle of reflection.

(What happens if the particle hits a corner is undefined, but the set of trajectories for which this happens is vanishingly small, i.e. has zero measure.)

We can ask the usual questions that dynamicists do when they probe the behavior of their systems: what do the orbits of this system look like? Are they periodic / closed, or dense? How does that depend on the initial data (i.e. position and direction)? What does the generic orbit look like? How many closed orbits are there?  Is the system ergodic? Mixing? If so, at what rate? How do the answers to all of these questions depend on P?

The great realm of irrational ignorance

Answering these questions can get fiendishly tricky surprisingly fast. For instance, very little is known in the case where the polygon P has angles which are not rational multiples of \pi.

Even more narrowly, and somewhat astonishingly, it is still an open question to determine if billiards on a general triangular P has a closed orbit. When P is acute, the triangular path formed by the line segments between the feet of the altitudes can be shown to be a billiards path (as demonstrated by Fagnano more than 240 years ago); when P is a right-angled there is a similarly explicit construction. What happens when P is obtuse is, still, anybody’s guess—although Rich Schwartz has at least partial, computed-assisted results.

Rational enlightenment via translation surfaces

By contrast, when the angles of P are all rational multiples of \pi, there is great deal that can be said, using tools from such varied fields as Riemann surfaces, Teichmüller theory, hyperbolic geometry, and even algebraic geometry.

What allows us to start applying all of these diverse tools is a relatively simple device, the translation surface associated to the billiard system, which might be seen as akin to development maps for (G,X)-structures—the rough idea in both cases being to unroll transitions until we see everything at once on a single object, or, in slightly more technical language, to globalize the coordinate charts.

We can define translation surfaces more generally, as maximal atlases of coordinate charts into the Euclidean plane \mathbb{R}^2, with finitely many cone singularities and where the transition maps between coordinate charts are translations in \mathbb{R}^2. Note that we can obtain the genus from information about the cone singularities via (a discrete version of) Gauss-Bonnet.

We may equivalently define translation surfaces—although to show the equivalence takes a little work—as surfaces built from some finite collection of polygons embedded in the Euclidean plane with a distinguished direction (and so inheriting an Euclidean metric and a distinguished direction—by gluing maps between their sides which are translations.

For a translation surface coming from a billiard table, we may take of the polygons to be congruent. In this sense these translation surfaces have additional symmetries that we should not expect a generic translation surface to have, and will likely be rather special in this sense (although there might conceivably be some sort of rigidity result lurking somewhere.)

Closed orbits: Masur’s theorem

The more general point of view of translation surfaces allows us more freedom of argument in proving things about our original, more restricted context.

For instance: a theorem of Masur states that periodic orbits exist on every translation surface; in fact, there are infinitely many. In fact, he proves even more: the number of periodic orbits of length N grows quadratically in N. If our translation surface did in fact come from a polygonal billiard system, these periodic orbits project down to closed trajectories.

The proof uses the idea of “changing the translation structure while preserving the affine structure”, as J. Smillie’s survey describes it:

Closed orbits can be detected geometrically via the presence of cylinders—subsets of our translation surface isometric to S^1 \times I. If we fix the genus, area, and singularity data for our surface, the translation surface can only have large diameter if it contains a (long) cylinder. In particular, if our translation surface had area greater than some universal constant D, we have our cylinder, and hence a closed orbit.

Otherwise, we observe that we have some freedom to change the translation structure, i.e. the coordinate charts and translations involved in the transitions—without changing the resulting affine geometry on the translation surface. Such a change takes cylinders to cylinders, but changes the metric geometry, and in particular the diameter. We can argue more carefully to find that there is always some such change of translation structure which takes the diameter to or beyond our universal constant D; and so the general case in fact reduces to our earlier, easier one.

(With a lot more care, we may also obtain the quadratic asymptotics stated above.)

In effect, the translation surface point of view tells us that polygonal billiards on families of polygons which produce the same translation surface have similar dynamical / geometric behavior—something which was not at all obvious just looking at the polygons themselves.

Ergodicity of the billiard

A dynamical system (X,T,\mu) is said to be ergodic (w.r.t. the given measure \mu) if any T-invariant subset of X has either zero or full measure w.r.t. \mu. Ergodic systems are, in some precise sense via the ergodic decomposition, the building blocks of all dynamical systems.

An example of an ergodic system is given by polygonal billiards on a rectangle, which is in fact equivalent to the geodesic flow on a flat torus. In fact, the flow / billiard trajectories in almost every direction are ergodic, and moreover equidistribute (this can then be shown to imply unique ergodicity—see below); the non-ergodic directions are precisely those with rational slopes, and are all periodic.

This especially nice state of affairs is a prototypical example of Veech dichotomy, which states that every direction for the constant-slope flow is either uniquely ergodic, or periodic. The former often form a small subset, but what “often” and “small” here entail precisely is still not entirely pinned down.

It follows from classical results that integrable polygons (whose corresponding translation surfaces are tori) satisfy Veech dichotomy. Gutkin proved that “almost-integrable” polygons also satisfy the dichotomy. Veech, in 1989, that all lattice surfaces (now also called Veech surfaces) satisfy the dicohotomy. Here, a lattice surface is one whose group of orientation-preserving affine automorphisms, or rather the image thereof under taking the derivative, forms a lattice in \mathrm{SL}(2,\mathbb{R}). All the previous examples are Veech surfaces, as well as those corresponding to regular n-gons, as well as certain triangles.

This is still not the most general class of translation surface satisfying Veech dichotomy, however: Smillie and Weiss proved in 2008 that there are non-lattice surfaces satisfying Veech dichotomy.

This result of Veech, plus the difficulty of the problem in its full generality, seems to have led to a focus on studying the ergodicity of the billiard flow in a fixed direction.

Unique ergodicity and minimality

(X,T,\mu) is uniquely ergodic if \mu is the only invariant probability measure—in this case, since any invariant measure is a convex linear combination of ergodic measures, \mu is necessarily ergodic.

The Bunimovich stadium is an example of a dynamical system—a planar billiard, in fact, although not polygonal—which is known to be ergodic but not uniquely ergodic.

Since the systems we are working with are naturally equipped with both a measure and a topology, we can also ask about minimality (a topological analogue of ergodicity—a minimal system has no proper closed T-invariant subsets), and how it relates to ergodicity.

It can be proven that a flow direction is minimal if there are no saddle connections (i.e. geodesic segments starting and ending at vertices, but not passing through any vertices in their interior) in that direction. The proof starts with the observation that it is useful to consider the first return map to a transverse interval in the surface. This turns out to be an interval exchange transformation (IET.) IETs are a well-studied class of dynamical systems, and criteria for the minimality of IETs lead to criteria for the minimality of directional flows, including this particular one.

From the combined / somewhat muddled viewpoint of dynamics both measurable and topological, minimal ergodic directions are especially nice, and non-minimal directions we might attempt to deal with using some sort of induction / topological reduction; so we might hope that there aren’t any minimal non-ergodic directions to deal with. It is a straightforward corollary of Veech dichotomy that Veech surfaces (or more generally surfaces which satisfy the dichotomy) do not have such directions.

However, Masur has produced examples of minimal non-ergodic directions for the geodesic flow on a translation surface of genus 2, by considering a translation structure given by a rectangular billiards table with two slits in the interior; these can be generalized to give (uncountably many) examples of minimal non-ergodic directions on any translation surface of genus 2 or greater.

Enter the Teichmüller geodesic

However, it can be proven that set of such directions has Lebesgue measure zero; in fact it can be proved that rational billiards are uniquely ergodic in almost every direction .

The key step is a result of Masur that if (X, \omega) is a translation surface for which flow in the vertical direction is not uniquely ergodic, then the Teichmüller geodesic associated to (X, \omega) is divergent, i.e. it eventually leaves every compact set in the moduli space.

Here the Teichmüller geodesic associated to (X, \omega) is the flowline of (X,\omega) under the diagonal subgroup \left\{ \left( \begin{array}{cc} e^t \\ & e^{-t} \end{array} \right) : t \in \mathbb{R} \right\} of \mathrm{SL}(2,\mathbb{R}) (the Teichmüller geodesic flow, for it really does correspond to flowing along geodesics in Teichmüller space), projected to moduli space.

There has been further work to figure out just “how big” the set of non-ergodic directions is, e.g. in terms of Hausdorff dimension.

Aside: smooth non-polygonal billiards

From a slightly different perspective, billiard trajectories are akin to geodesic flow trajectories in the plane. With this in mind, perhaps the following result is not too surprising:

Any smooth surface in 3-space may be flattened to obtain something close to a smooth (not necessarily polygonal) billiard table in 2-space. Kourganoff showed that, under some mild hypotheses, the geodesic flow of this surface converges locally uniformly to the billiard flow. Moreover, if the billiard is dispersive (i.e. any two distinct points have neighborhoods whose orbits are eventually separated) and has finite horizon (i.e. time between collisions remains bounded), then the geodesic flow of the corresponding surface is Anosov. This result can be applied to the theory of mechanical linkages and their dynamics, to provide e.g. novel examples of simple linkages whose physical behavior is Anosov.

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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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Arguments using moduli spaces: some examples

This blogpost started as a response to the question “why moduli spaces?” … but then David Ben-Zvi had a good pithy answer to that in his  Princeton Companion to Mathematics article on moduli spaces : “Moduli spaces can be thought of as geometric solutions to geometric classification problems.”

If the geometry of the moduli space is well-chosen, it can tell us things about the geometry of the spaces / objects which points in it correspond to. Ben-Zvi has a thoroughly concrete and persuasive discussion of this in the case of 1-dimensional real projective space, a.k.a. the moduli space of lines through the origin in the plane.

Below we document, somewhat briefly, a couple more examples of how we can say things properties of geometric objects of a certain type by arguing in a moduli space of all such objects.

An example from (classical) algebraic geometry

A conic (or conic section) is a curve obtained by the intersection of a cone with a plane; familiar examples, known to the Greeks, include circles, ellipses, and hyperbolae.

Conics in 2-dimensional (complex) projective space \mathbb{P}^2 = \mathbb{P}(\mathbb{C}^3) are parametrized by \mathbb{P}(\mathrm{Sym}^2 k^3) \cong \mathbb{P}^5; explicitly, we have the correspondence [a:b:c:d:e] \leftrightarrow ax^2 + bxy + cy^2 + dyz + ez^2 = 0.

In other words, \mathbb{P}^5 is the moduli space of conics in \mathbb{P}^2. We will use this to show that given any five distinct points (p_1, p_2, p_3, p_4, p_5) in \mathbb{P}^5 such that no four lie on a (projective) line, there is a unique conic (possibly degenerate) passing through (p_1, p_2, p_3, p_4, p_5).

The conics through any given point in \mathbb{P}^2 are parametrized by a hyperplane in the moduli space \mathbb{P}^5; conics through 5 distinct points are given by the intersection of five hyperplanes, which is non-empty by considerations of dimension. This already gives us existence, even without the “no four on a line” condition; it remains only to show uniqueness given this additional condition.

Suppose this intersection consists of a single point (in \mathbb{P}^5, i.e. a unique conic in \mathbb{P}^2.) Then it is clear that we cannot have any four of our p_i on a line \ell, for taking the union of this line and a line between the remaining point and any one of the four yields a conic containing the five given points, contradicting uniqueness.

Conversely, suppose the intersection is larger, i.e. it contains a \mathbb{P}^1 (a “pencil”) of conics.

We parametrize this pencil by \{sF + tG | [s:t] \in \mathbb{P}^1\} for some F, G. Then, for any q \in \mathbb{P}^2 on the line containing p_i, p_j, sF(q) + tG(q) = 0 has a solution (in s, t), i.e. there is a conic in our pencil going through q. We may verify that any conic in \mathbb{P}^2 containing three collinear points is a pair of lines (or a double line); by the pigeonhole principle, at least one of these lines \ell must contain at least three of the p_i.

Now if \ell contains exactly three of the p_i, then there is a unique conic containing all five p_i—the pair of lines consisting of \ell and the line through the remaining two p_i. Since by assumption we do not have uniqueness, \ell must contain (at least) four of the p_i.

An example from geometric topology

The genus g Teichmüller space Teich(g) is the space of all complex / conformal structures on a closed Riemann surface of genus g, up to some natural notion of equivalence. Equivalently, via uniformization, it is a space of all constant-curvature Riemannian metrics on a topological surface of genus g.

We will focus on the case of surfaces of genus at least two; for these surfaces, uniformization tells us that the constant-curvature metrics are hyperbolic. There is an analogous theory for genus one which gives us the moduli space of flat metrics on the torus, and many of the same general arguments that appear below can be applied there as well, but the specifics there can have a slightly different flavour, due to the lack of negative curvature (in the objects, that is.)

A point in Teich(g) can be specified by a pair (S, h), where S is a surface with a hyperbolic metric, and h: \pi_1(F) \to \pi_1(S) is a marking, a choice of isomorphism from the fundamental group of a “naked” topological (reference) surface F without a metric and our metrized surface S. To steal Dick Canary’s metaphor: S is the hyperbolic clothing, and h provides instructions for how to wear it.

Technically, a point in Teich(g) is an equivalence class of such pairs, where two marked hyperbolic metrics (S_1, h_1) and (S_2, h_2) are considered equivalent if there is an isometry j: S_1 \to S_2 such that j \circ h_1 \simeq h_2, i.e. the two maps are homotopic to each other. In terms of the metaphor: jiggling the clothing around a little doesn’t give an essentially different way of wearing it.

We have specified Teich(g) as a set. We can give it a topology in (at least) two ways:

  1. By identifying each equivalence class of marked hyperbolic structures [(S,h)] with a holonomy representation h_*: \pi_1(F) \to \mathrm{Isom}^+(\mathbb{H}^2) \cong \mathrm{PSL}(2,\mathbb{R}) and thus Teich(g) with a subspace of the representation variety X_2(F) := \mathrm{Hom}(\pi_1(F), \mathrm{PSL}(2,\mathbb{R})) / \mathrm{PSL}(2,\mathbb{R}), where the quotient identifies representations that are conjugate in \mathrm{PSL}(2,\mathbb{R})—this corresponds in the algebra to the identification of homotopic marked metrics in the geometry.X_2(F) isn’t actually a variety, in the algebraic geometry sense, but it does have natural topology. The induced topology on Teich(g) is called the algebraic topology.
  2. By defining a metric on Teich(g), the Teichmüller metric d_{Teich}, and giving it the metric topology. d_{Teich}([(S_1, h_1)], [(S_2, h_2)]) is defined as (the logarithm of) the minimum quasiconformal distortion over all quasiconformal maps between the two marked hyperbolic structures.(For full definitions see e.g. Chapter 11 of Farb and Margalit’s Primer on Mapping Class Groups.)

A moment’s thought and some fiddling shows that these two approaches produce the same topology.

Intuitively, neighborhoods in this topology correspond to sets of marked hyperbolic metrics which don’t differ from each other too much—the clothing has similar dimensions, and the instructions are relatively similar.

A space to act on

The mapping class group Mod(F) of a surface (or more generally of any space) F is defined as the group of all isotopy classes of homeomorphisms from to itself. Mod(F) may be thought of as the group of symmetries of a surface. The notation Mod comes by way of analogy with SL(2, Z), which is the mapping class group of the torus—the mapping class group for a surface is sometimes also called the Teichmüller modular group.

The mapping class group of a genus g surface acts naturally on the genus g Teichmüller space by change of marking: \varphi \cdot [(S,h)] = [(S, \varphi_* h)], i.e. by changing the layout of the naked topological surface, it yields effectively different instructions for how to wear the hyperbolic clothing.

It can be shown that this action is properly discontinuous (see e.g. section 12.3 of Farb and Margalit), and isometric if we give Teich(g) the Teichmüller metric. The quotient of Teich(g) is \mathcal{M}_g, the moduli space of genus g surfaces—which we discuss slightly more below,

Using this action, we may obtain Thurston’s classification of elements of Mod(S), analogous to classification of elements of SL(2, R) by their action on hyperbolic 2-space: define the translation length \tau(\phi) := \inf_{\mu \in \mathrm{Teich}(g)} d_{Teich}(\mu, \phi(\mu)); then

  • periodic elements \phi have fixed points in Teichmüller space, and hence have \tau(\phi) = 0 and achieve this inf (analogous to elliptic elements of SL(2, R));
  • pseudo-Anosov elements \phi have positive translation length, and always achieve this translation length on a Teichmüller geodesic (analogous to hyperbolic elements of SL(2, R));
  • reducible elements \phi have zero translation length, but do not achieve it (analogous to parabolics.)

A fuller description of this classification and the work needed to obtain it can be found in Chapter 14 of Farb and Margalit’s Primer.

A space to explore

It would perhaps been possible to rephrase the above classification and obtain it without the use of Teichmüller space, but the use of Teichmüller space certainly helps illuminate the (geometric) structure of the argument and how it is analogous to the classification of elements of SL(2, R) essentially using hyperbolic geometry.

Similarly, moduli spaces can provide a new light in which to consider / describe possible geometric structures.

For instance, we might ask: how could we imagine or describe the various hyperbolic metrics can we put on a genus g surface?

One possible approach to answering this could involve putting coordinates on the genus g Teichmüller space and attempting to interpret those coordinates in terms of the genus 2 surface. Indeed, we can put a set of (global!) 6– 6 coordinates on Teichmüller space, the Fenchel-Nielsen coordinates, which we obtain by considering a set of 3g – 3 disjoint geodesics on our surface (a pants decomposition, so called because it divides our surface into spheres with three boundary components, i.e.  pairs of pants) and taking length and twist parameters along those geodesics—for a full description see Danny Calegari’s post (linked right above).

The Fenchel-Nielsen coordinates (together with a little hyperbolic geometry) give us a very concrete way of answering our question: to obtain some hyperbolic metric on a genus 2 surface, say, we

  1. take two pairs of pants,
  2. identify pairs of boundary curves that we will glue to obtain our genus 2 surface,
  3. specify how long we want each pair of boundary curves to be—this uniquely determines the hyperbolic metric on each pair of pants—,
  4. specify how much we twist the curves in each pair relative to each other when we glue them together.

To obtain a different metric, we change our specifications for the lengths and/or the twist parameters.

Or, jumping up a dimension, we might ask: are there any hyperbolic metrics on closed 3-manifolds that we could describe similarly concretely?

One way to answer this would be to consider a corresponding representation variety \mathrm{Hom}(\pi_1(M), \mathrm{PSL}(2,\mathbb{C})) // \mathrm{PSL}(2,\mathbb{C}). Unfortunately, 3-manifold fundamental groups are much more complicated than surface groups, and these representation varieties have been much harder to study (although, using machinery I do not [yet?] understand, they are actually algebraic varieties.)

Deformation, degeneration, and the totality

Putting the totality of geometric objects together as a moduli space highlights ways of varying the geometry on them which may be less obvious or natural from a non-moduli-space perspective:

  • Infinitesimal deformations of a particular geometric structure: e.g. we might ask, how many degrees of freedom are there if we start with a given hyperbolic metric and try to vary the lengths of the geodesics? Can we concretely describe or characterize these degrees of freedom? Barry Mazur has an excellent overview article which talks more about deformations.
  • Degeneration of particular aspect/s of our geometric structures—e.g. what happens to our hyperbolic metric if we pick a geodesic loop and shrink its length towards zero, or expand it towards infinity?

Having such a structured totality also allows us to answer with some degree of precision questions such as “how special is such a feature (say, the presence of many short disjoint geodesics) for this type of geometric structure?” With a moduli space in hand, we might be able to answer “it is a high-codimension feature” (so rather special), or perhaps “it is true except in small balls of finite total volume” (so not very special.)

Aside: coarse and fine moduli; universal properties

The distinction between Teichmüller space Teich(g) and its quotient moduli space \mathcal{M}_g, briefly mentioned above, more generally reflects a distinction between fine and coarse moduli spaces. The emphasis on marked hyperbolic metrics may not seem all that natural: we may not care so much about different instructions for how to wear the clothing, but are only interested in essentially different sorts of hyperbolic clothing—which is, untranslating the metaphor, what \mathcal{M}_g is describing.

\mathcal{M}_g, though, is not as nice as Teich(g)—for one thing, Teich(g) has the universal property that any continuously-varying family of marked genus g surfaces parametrized by a topological space S is described by a continuous map from S into Teich(g). Indeed, we can put a complex structure on Teich(g), and then we can replace “continuous” above with “complex analytic”.

\mathcal{M}_g, however, does not have this property, essentially because any higher-genus surface has non-trivial automorphisms, which kill any hope of obtaining the universal property; to address this problem, we “rigidify” the surface (or, from another perspective, “kill off the automorphisms”) by specifying a marking.

Such a universal property provides one way of specifying, categorically, what a moduli space is, and is one way of more precisely expressing the notion that the moduli space captures the geometry that we are interested in.

For more on this circle of ideas—really quite key to the history of Teichmüller space—, introduced by Teichmüller in his 1944 paper and expanded upon by Grothendieck in a series of lectures at the Séminaire Henri Cartan, see this article of A’Campo, Ji, and Παπαδόπουλος.

Invariants from moduli space

Moduli spaces also allow for a novel way of defining topological invariants, which, again, Ben-Zvi’s article describes much better than I could:

‘an important application of moduli spaces in geometry and topology is inspired by quantum field theory, where a particle, rather than follow the “best” classical path between two points, follows all paths with varying probabilities. Classically, one calculates many topological invariants by picking a geometric structure (such as a metric) on a space, calculating some quantity using this structure, and finally proving that the result of the calculation did not depend on the structure we chose. The new alternative is to look at all such geometric structures, and integrate some quantity over the space of all choices. The result, if we can show convergence, will manifestly not depend on any choices. String theory has given rise to many important applications of this idea, in particular by giving a rich structure to the collection of integrals obtained in this way. Donaldson and Seiberg-Witten theories use this philosophy to give topological invariants of four-manifolds. Gromov-Witten theory applies it to the topology of symplectic manifolds, and to counting problems in algebraic geometry.’

We remark here that Vassiliev invariants in knot theory can be seen as another example of this approach; this is perhaps clearer from the point of view of the Kontsevich integral, rather than from the combinatorial perspective.

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Two (co)compactness criteria for lattices

Here and H are Lie groups, \Gamma and \Lambda are lattices in one of those Lie groups, and C is a subset of the Lie group.

Detecting (non)compactness via accumulation at the identity

Conjugates by compact sets do not accumulate: The image of C \subset H in H / \Lambda is precompact iff the identity element \mathrm{id}_H is not an accumulation point of the conjugated lattice {}^C\Lambda.

[[ some pictures to be inserted here ]]

Proof: {}^C\Lambda \cap U^{-1}U = \{e\} iff we cannot find distinct u_1, u_2 \in U s.t. u_1c \Gamma = u_2c \Gamma for all c \in C, iff the translated orbit maps (covering maps!) H \to H/\Lambda given by h \mapsto hc\Lambda are injective on U for every c \in C.

Now if our image is precompact, then we may cover it by finitely many pancakes (i.e. neighborhoods in H / \Lambda diffeomorphic to each connected component of their preimage in the cover H); by this finiteness, and H-equivariance, we can find some small open neighborhood U so that the desired injectivity condition on our translated orbit maps holds.

If our image is not precompact, then given any small open neighborhood U we can find a sequence of elements in C escaping any compact translate U^{-1}K. By an inductive / diagonal argument (involving larger and larger compact sets, each of which contains a previous Uc_n\Lambda) we have a sequence (c_n) \subset C with Uc_n\Lambda pairwise disjoint. Since \Lambda has finite covolume, these cannot all have the same (nonzero) volume, and so (by fundamental domain considerations) not all of the translated orbit maps are injective on U.

Mahler’s compactness criterion is the special case where H = \mathrm{SL}(\ell,\mathbb{R}) and \Lambda = \mathrm{SL}(\ell,\mathbb{Z}).

This can be proven without using that \Lambda is a lattice, by considering \mathrm{SL}(\ell,\mathbb{R}) / \mathrm{SL}(\ell,\mathbb{Z}) as the moduli space of unit-covolume lattices in \mathbb{R}^\ell (or: “wow, the lattices are back at a whole new level!”): then the result says that a set of unit-covolume lattices is compact iff it does not contain arbitrarily short vectors.

The forward implication is fairly immediate after a moment’s thought; the reverse implication follows by considering the shortest vectors which appear in the lattices of any given sequence from our set, observing that those vectors form bounded orbits, taking (sub)sequential limits in those orbits using compactness in \mathbb{R}^\ell, and finally passing back to (sub)sequential limits in the sequence of lattices.

Detecting (non)compactness with unipotents

Cocompact lattices have no unipotents: The homogeneous space G / \Gamma is compact only if \Gamma has no nontrivial unipotents.

One proof [of the contrapositive] follows from (the general version of) Mahler’s criterion above, via the Jacobson-Morosov Lemma which gives a map \phi: \mathrm{SL}(2,\mathbb{R}) \to G sending our favorite unipotent \left( \begin{array}{cc} 1 & 1 \\ & 1 \end{array} \right) to a nontrivial unipotent u \in \Gamma. Call a the image under phi of our favorite hyperbolic element \left( \begin{array}{cc} 2 \\ & 1/2 \end{array}\right). Then a^nua^{-n} accumulates to the identity, and so G / \Gamma is not precompact by the first criterion above.

More is true if we have more structure on our lattices:

Godement: Suppose G is defined over a real algebraic number field F (e.g. \mathbb{Q}), and let \mathcal{O} be the ring of integers in F. The homogeneous space G / G_{\mathcal{O}} (in the case of F = \mathbb{Q}, G / G_{\mathbb{Z}}) is compact iff G_{\mathbb{Z}} has no nontrivial unipotents.

[[ some more pictures here ]]

Proof of the new (reverse) direction uses the arithmetic structure on G_{\mathbb{Z}}. Again we prove the contrapositive. Suppose G / G_{\mathbb{Z}} is not compact. By (Mumford’s generalisation of) the Mahler criterion above, this implies {}^gG_{\mathbb{Z}} accumulates at the identity for some g \in G. By continuity, the characteristic polynomials of the elements in an accumulating sequence g\gamma_n g^{-1} converge (in the sense of coefficients; this makes sense if we are working within some fixed ambient \mathrm{SL}(\ell, \mathbb{R}), for example) to (x-1)^\ell.

But this implies, by matrix similarity, that the characteristic polynomials of the \gamma_n converge to the same; and now we use the fact that \gamma is an integer matrix to say that this implies the \gamma_n, for n sufficiently large, has characteristic polynomial (x-1)^\ell, and hence is unipotent.

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The many faces of the hyperbolic plane

\mathbb{H}^2 is the unique (up to isometry) complete simply-connected 2-dimensional Riemann manifold of constant sectional curvature -1.

  1. It is diffeomorphic to \mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2) as a topological space (or, indeed, isometric as a Riemannian manifold, if we give \mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2) a left-invariant metric): to show this, we note that \mathbb{H}^2 has isometry group \mathrm{SL}(2,\mathbb{R}) / \pm\mathrm{id}, and the subgroup of isometries which stabilize any given p \in \mathbb{H}^2 is isomorphic to \mathrm{SO}(2).
  2. Since any positive-definite binary quadratic form is given by a symmetric 2-by-2 matrix with positive eigenvalues, and since the group of linear transformations on \mathbb{R}^2 preserving any such given form is isomorphic to \mathrm{O}(2), \mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2) is also the space of positive-definite binary quadratic forms of determinant 1, via the map from \mathrm{SL}(2,\mathbb{R}) to the symmetric positive-definite 2-by-2 matrices given by g \mapsto g^T g.
  3. \mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2) \cong \mathbb{H}^2 is also the moduli space of marked Riemann surfaces of genus 1, i.e. the Teichmüller space Teich(S) of the torus S. One way to prove this is to note that any such marked surface is the quotient of \mathbb{R}^2 by a \mathbb{Z}^2 action; after a suitable conformal transformation, we may assume that the generators of this \mathbb{Z}^2 act as z \mapsto z + 1 and z \mapsto z + \tau for some z \in \mathbb{H}^2 (in the upper half-plane model.) But now \tau is the unique invariant specifying this point in our moduli space.
  4. Since any marked Riemann surface of genus 1 has a unique flat metric (inherited as a quotient manifold of the Euclidean plane), \mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2) is also the moduli space of marked flat 2-tori of unit area.
  5. Since there is a unique unit-covolume marked lattice associated to each marked complex torus in the above, \mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2) is also the space of marked lattices in \mathbb{R}^2 with unit covolume.
  6. Note we may go directly between marked Riemann surfaces and quadratic forms by considering intersection forms.
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