Articles

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.)

Standard

A virtually free RACG

(Thanks to Harry Richman for bringing this example up.)

The free group $\langle a, b \rangle$ and the right-angled Coxeter group $\rangle a, b, c, d | a^2, b^2, c^2, d^2 \rangle$ have the same Cayley graph—the infinite 4-valent tree—; hence they are quasi-isometric. We note here that quasi-isometries are not required to group homomorphisms, and indeed the one we have here—an “identity map” of sorts (in an admittedly very loose and imprecise sense) between Cayley graphs—is not.

By results of Stallings and Dunwoody involving ends of groups ad accessibility (although Chapter 20 of Drutu-Kapovich’s draft manuscript seems to be the only reference I can find for this online), free groups are quasi-isometrically rigid, and hence, since finite-index subgroups of free groups are free, the RACG is virtually free (!)

With a little more thought we can explicitly exhibit a finite-index free subgroup: the subgroup of the RACG generated by ab, ac and ad is a nonabelian free group on three generators; we can verify it has finite index via a covering space argument.

Now I wonder if there is a finite-index $F_2$-subgroup of the RACG as well, so that the commensurability need not involve passing to finite-index subgroups on both sides. It seems like there shouldn’t be, although I can’t quite prove this yet.

Standard
Theorems

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.)

Standard
Articles

Geometric group theory (II)

Groups from geometry

Besides groups which have geometry, there are also groups with various geometric origins, and these make up another broad vein of the federated entity that is geometric group theory. Often these geometric origins allow us to give the groups themselves a certain geometry; but they also give us directly geometric tools for studying these groups.

The first examples we might think of include surface groups and hyperbolic 3-manifold groups, and more generally fundamental groups of geometric objects. And then we start thinking more …

Mapping class groups

The mapping class group Mod(S) of a surface S of finite type is the group of all homeomorphisms from the surface to itself, mod homotopy. These homeomorphisms are required to fix the boundary pointwise, and preserve the set of punctures setwise (i.e. mapping classes may permute punctures.)

The notation originates with Fricke, who called the “automorphic modular group”, by analogy with the classical modular group $\mathrm{SL}_2\mathbb{Z}$ which indeed is the mapping class group of the torus.

Mapping classes represent essential symmetries of the topological surface. Mapping class groups can be computed, in general, using the Alexander method—see e.g. Section 2.3 of Farb and Margalit’s Primer.

A large class of easily visualizable mapping classes is given by Dehn twists. There are many mapping classes which are not Dehn twists, but it turns out that mapping class groups are generated by a finite number of (judiciously-chosen) Dehn twists.

One proof of this proceeds by looking at the action of the mapping class group on a (slightly-modified) curve complex. The mapping class group acts on Teichmüller space by changing the marking—the mapping class $\varphi$ sends the point $[(S,h)]$ to $[(S, h \circ \varphi)]$—and also on the curve complex and its relatives, by extending the obvious action on the 1-skeleton (curves go to curves, and we can check that adjacency relations in the curve complex are preserved.)

Looking at these two group actions also gives us proofs that mapping class groups are finitely-presented, residually finite, satisfy a subgroup alternative, and so on.

Note that these two actions complement each other, in the sense that Teichmüller space is compact(ifiable) but not (coarsely) hyperbolic, whereas the curve complex is not locally-compact but is coarsely hyperbolic. Playing these off against each other is key in one proof of the subgroup alternative, for instance.

Outer automorphism groups of free groups

Given any group $\Gamma$ in general, the outer automorphism group $\mathrm{Out}(\Gamma)$ consists of all the automorphisms of $\Gamma$, modulo inner automorphisms.

Outer automorphism groups $\mathrm{Out}(F_n)$ of (finitely-generated, non-abelian) free groups $F_n$ have a particularly geometric flavour, since they may be thought of in terms of their action on a graph with fundamental group $F_n$, i.e. (or e.g.) a bouquet of n circles.

This leads us to the construction of Culler-Vogtmann outer space $CV_n$, a space of marked metric graphs, whose points are specified by pairs $[(X, h)]$ where X is a metric graph with fundamental group (isomorphic to) $F_n$ and $h: F_n \to \pi_1(X)$ is an isomorphism of fundamental groups, and the square braces indicate that we are quotienting out by homotopy equivalences.

This appears very analogous to the definition of Teichmüller space, and indeed there is an analogy between Teichmüller theory and the theory of $\mathrm{Out}(F_n)$ which, although somewhat informal and imperfect, has driven and illuminated much of the development of the latter.

Similar perspectives may be brought to bear on the study of $\mathrm{Out}(\Gamma)$ wherever $\Gamma$ is naturally the fundamental group of some nice class of geometric or topological objects, e.g. if $\Gamma$ is a RAAG (which is the fundamental group of its associated Salvetti complexes.)

Coxeter groups and Artin groups

(Most of what appears in this section was taken from either Ruth Charney’s survey notes, or these notes of Luis Paris.)

Coxeter groups are groups which generalize discrete reflection groups. The finite Coxeter groups are precisely the finite Euclidean reflection groups, and indeed any Coxeter group may be realized as a discrete reflection group

Coxeter groups have presentations of the form $\left\langle s_1, \dots, s_n | s_i = s_i^{-1}, s_i s_j s_i \cdots = s_j s_i s_j \cdots \right\rangle$, where relations of the second sort equal numbers of generators (alternating between $s_i$ and $s_j$) on both sides. The information needed to write this representation (or a representation for the corresponding Artin group, see below) may be encoded in the form of a finite graph, known as Dynkin diagrams.

Artin groups are also known as generalized braid groups, a braid group being (one way to define it being) the mapping class group of a disk with finitely many punctures. Braid groups have presentations of the form $\left\langle \tau_1, \dots, \tau_n | \tau_i \tau_j = \tau_j \tau_i, \tau_i \tau_{i+1} \tau_i = \tau_{i+1} \tau_i \tau_{i+1} \right\rangle$. More generalized Artin groups are given by presentations of the form $\left\langle s_1, \dots, s_n | s_i s_j s_i \cdots = s_j s_i s_j \cdots \right\rangle$, where the relations are of the same form as those which appear above in presentations for Coxeter groups.

We note that these presentations are very similar—the only difference being the involutive relations for the generators in a Coxeter group presentation—, and indeed to every Coxeter group there is an associated Artin group (and vice versa) obtained by removing (or adding, resp.) those involutive relations.

Proceeding along this line of thought leads to a geometric interpretation of Artin groups: whereas a Coxeter group is a group of reflections in some finite hyperplane arrangement, the corresponding Artin group A is the fundamental group of the corresponding complexified hyperplane arrangement $Y_W$ modulo the action of the Coxeter group. Moreover, at least for W finite, $Y_W / W$ is aspherical, so it is in fact a $K(pi, 1)$; the same is conjectured to be true for more general W,

Artin groups associated to finite Coxeter groups (called spherical Artin groups) sharmany properties of braid groups. Non-spherical Artin groups (i.e. those whose associated Coxeter groups are infinite) are much less well-understood, but are conjectured to share many of the same good algorithmic / geometric properties of their spherical cousins.

Lie groups and lattices

A different class of examples comes from remarking that fundamental groups of closed manifolds are lattices in Lie groups, and asking about more general lattices in Lie groups.

This allows us to leverage (algebraic) tools from Lie theory to use alongside our geometric tools to study our examples.

[[ future addition: examples in lower rank / rigidity and superrigidity results in higher rank ]]

Dynamical perspectives: Property (T) and amenability

A third major vein of the not-quite-a-single-subject involves using dynamics to study group actions and what they say about groups. Here (at least) two things stand out.

One is Kazhdan’s property (T): a group has (T) if any unitary representation of a group which has an almost-invariant vector has an invariant vector.  Equivalently, [[long list here]]

[[ future addition: equivalent characterizations of (T) and properties of (T) groups ]]

Kazhdan proved that lattices in Lie groups have (T) iff the ambient Lie groups have (T). Simple Lie groups of higher rank have (T), and thus for $n \geq 3$, $\mathrm{SL}_n\mathbb{Z}$ has (T). His motivation for all of this was apparently to demonstrate the finite generation of these lattices (it can be shown that any discrete countable group with (T) is finitely-generated.)

It seems a bit of a crazy detour, but perhaps entirely reasonable if one is well-versed and comfortable in the language of unitary representations.

On the other hand, there is amenability: [[ future addition: (at least part of) long list of equivalent definitions ]].

Amenable groups are, from a dynamical perspective at least, similar to abelian groups—many dynamical arguments that work for abelian groups can be made to work, in some form, for amenable groups.

[[ future addition: some  properties of amenable groups ]]

Amenable groups with property (T) are compact, and in some sense amenability and (T) are opposite: amenability makes almost-invariant vectors easy to find, whereas (T) makes them hard to find.

Standard
Annotated Definitions

Proper discontinuity

(Credit where due: this is a summary things from MathOverflow 55726 and Math Stack Exchange 1082834.)

Let G be a locally-compact group acting on a locally-compact Hausdorff topological space X (these additional conditions on G and X being certainly satisfied if e.g. G is finitely-generated and X is a proper metric space.)

We say the action is properly discontinuous if for any compact set $K \subset X$, $K \cap gK \neq \varnothing$ for at most finitely many $g \in G$. (cf. the action of $\mathbb{Z}^n$ on $\mathbb{R}^n$ by translations, which is discontinuous: for any [sufficiently small] compact set K$K \cap gK = \varnothing$ for every $g \in \mathbb{Z}^n$; here we allow finite overlap even at arbitrarily small resolution: this allows for finite stabilizers, parabolic elements, etc.)

In particular this implies that every point $x \in X$ has an open neighborhood U s.t. $U \cap gU \neq \varnothing$ for at most finitely many $g\ in G$. (Take U to sit inside some compact set K.)

Note that the second condition is strictly weaker than the first: consider the $\mathbb{Z}$-action on the punctured plane given by $n \cdot (x,y) = (2^n x, 2^{-n}y)$. This satisfies the second condition, but not the first—take K to be any closed annulus centered about the origin, for instance.

The main point of imposing these mildly technical conditions when we study “geometric” group actions is this:

This second condition is equivalent to the quotient map $X \to X/G$ being a covering map. The first condition (the definition) serves to ensure that the map $G \times X \to X \times X$ given by $(g,x) \mapsto (x,gx)$ is proper, and hence that the quotient X/G is Hausdorff.

Standard