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One of the nice things about living in Hyde Park is the proximity to the University of Chicago. Consequently, over the summer I came in to the department from time to time to work in my office, where I have all my math books, fast internet connection, etc. One day in early September (note: the Chicago quarter doesn’t start until October, so technically this was still “summer”) I happened to run in to Volodya Drinfeld in the hall, and he asked me what I knew about fundamental groups of (complex) projective varieties. I answered that I knew very little, but that what I did know (by hearsay) was that the most significant known restrictions on fundamental groups of projective varieties arise simply from the fact that such manifolds admit a Kähler structure, and that as far as anyone knows, the class of fundamental groups of projective varieties, and of Kähler manifolds, is the same.

Based on this brief interaction, Volodya asked me to give a talk on the subject in the Geometric Langlands seminar. On the face of it, this was a ridiculous request, in a department that contains Kevin Corlette and Madhav Nori, both of whom are world experts on the subject of fundamental groups of Kähler manifolds. But I agreed to the request, on the basis that I (at least!) would get a lot out of preparing for the talk, even if nobody else did.

Anyway, I ended up giving two talks for a total of about 5 hours in the seminar in successive weeks, and in the course of preparing for these talks I learned a lot about fundamental groups of Kähler manifolds. Most of the standard accounts of this material are aimed at people whose background is quite far from mine; so I thought it would be useful to describe, in a leisurely fashion, and in terms that I find more comfortable, some elements of this theory over the course of a few blog posts, starting with this one.

I recently learned from Jim Carlson’s blog of the passing of Harsh Pittie on January 16 this year. I never met Harsh, but he is very familiar to me through his work, in particular for his classic book Characteristic classes of foliations. I first encountered this book as a graduate student, late last millenium. The book has several features which make it very attractive. First of all, it is short — only 105 pages. My strong instinct is always to read books cover-to-cover from start to finish, and the motivated reader can go through Harsh’s book in its entirety in maybe 4 hours. Second of all, the exposition is quite lovely: an excellent balance is struck between providing enough details to meet the demands of rigor, and sustaining the arc of an idea so that the reader can get the point. Third of all, the book achieves a difficult synthesis of two “opposing” points of view, namely the (differential-)geometric and the algebraic. In fact, this achievement is noted and praised in the MathSciNet review of the book (linked above) by Dmitri Fuks.

The purpose of this blog post is to try to give some insight into the “meaning” of the Hall-Witt identity in group theory. This identity can look quite mysterious in its algebraic form, but there are several ways of describing it geometrically which are more natural and easier to understand.

If $G$ is a group, and $a,b$ are elements of $G$, the commutator of $a$ and $b$ (denoted $[a,b]$) is the expression $aba^{-1}b^{-1}$ (note: algebraists tend to use the convention that $[a,b]=a^{-1}b^{-1}ab$ instead). Commutators (as their name suggests) measure the failure of a pair of elements to commute, in the sense that $ab=[a,b]ba$. Since $[a,b]^c = [a^c,b^c]$, the property of being a commutator is invariant under conjugation (here the superscript $c$ means conjugation by $c$; i.e. $a^c:=cac^{-1}$; again, the algebraists use the opposite convention).

I am Alden, one of Danny’s students. Error/naivete that may (will) be found here is mine. In these posts, I will attempt to give notes from Danny’s class on hyperbolic geometry (157b). This first post covers some models for hyperbolic space.

1. Models

We have a very good natural geometric understanding of ${\mathbb{E}^3}$, i.e. 3-space with the euclidean metric. Pretty much all of our geometric and topological intuition about manifolds (Riemannian or not) comes from finding some reasonable way to embed or immerse them (perhaps locally) in ${\mathbb{E}^3}$. Let us look at some examples of 2-manifolds.

• Example (curvature = 1) ${S^2}$ with its standard metric embeds in ${\mathbb{E}^2}$; moreover, any isometry of ${S^2}$ is the restriction of (exactly one) isometry of the ambient space (this group of isometries being ${SO(3)}$). We could not ask for anything more from an embedding.
• Example (curvature = 0) Planes embed similarly.
• Example (curvature = -1) The pseudosphere gives an example of an isometric embedding of a manifold with constant curvature -1. Consider a person standing in the plane at the origin. The person holds a string attached to a rock at ${(0,1)}$, and they proceed to walk due east dragging the rock behind them. The movement of the rock is always straight towards the person, and its distance is always 1 (the string does not stretch). The line traced out by the rock is a tractrix. Draw a right triangle with hypotenuse the tangent line to the curve and vertical side a vertical line to the ${x}$-axis. The bottom has length ${\sqrt{1-y^2}}$, which shows that the tractrix is the solution to the differential equation$\displaystyle \frac{-y}{\sqrt{1-y^2}} = \frac{dy}{dx}$

The Tractrix

The surface of revolution about the ${x}$-axis is the pseudosphere, an isometric embedding of a surface of constant curvature -1. Like the sphere, there are some isometries of the pseudosphere that we can understand as isometries of ${\mathbb{E}^3}$, namely rotations about the ${x}$-axis. However, there are lots of isometries which do not extend, so this embeddeding does not serve us all that well.

• Example (hyperbolic space) By the Nash embedding theorem, there is a ${\mathcal{C}^1}$ immersion of ${\mathbb{H}^2}$ in ${\mathbb{E}^3}$, but by Hilbert, there is no ${\mathcal{C}^2}$ immersion of any complete hyperbolic surface.That last example is the important one to consider when thinking about hypobolic spaces. Intuitively, manifolds with negative curvature have a hard time fitting in euclidean space because volume grows too fast — there is not enough room for them. The solution is to find (local, or global in the case of ${\mathbb{H}^2}$) models for hyperbolic manfolds such that the geometry is distorted from the usual euclidean geometry, but the isometries of the space are clear.

2. 1-Dimensional Models for Hyperbolic Space

While studying 1-dimensional hyperbolic space might seem simplistic, there are nice models such that higher dimensions are simple generalizations of the 1-dimensional case, and we have such a dimensional advantage that our understanding is relatively easy.

2.1. Hyperboloid Model

Parameterizing ${H}$

Consider the quadratic form ${\langle \cdot, \cdot \rangle_H}$ on ${\mathbb{R}^2}$ defined by ${\langle v, w \rangle_A = \langle v, w \rangle_H = v^TAw}$, where ${A = \left[ \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right]}$. This doesn’t give a norm, since ${A}$ is not positive definite, but we can still ask for the set of points ${v}$ with ${\langle v, v \rangle_H = -1}$. This is (both sheets of) the hyperbola ${x^2-y^2 = -1}$. Let ${H}$ be the upper sheet of the hyperbola. This will be 1-dimensional hyperbolic space.

For any ${n\times n}$ matrix ${B}$, let ${O(B) = \{ M \in \mathrm{Mat}(n,\mathbb{R}) \, | \, \langle v, w \rangle_B = \langle Mv, Mw \rangle_B \}}$. That is, matrices which preserve the form given by ${A}$. The condition is equivalent to requiring that ${M^TBM = B}$. Notice that if we let ${B}$ be the identity matrix, we would get the regular orthogonal group. We define ${O(p,q) = O(B)}$, where ${B}$ has ${p}$ positive eigenvalues and ${q}$ negative eigenvalues. Thus ${O(1,1) = O(A)}$. We similarly define ${SO(1,1)}$ to be matricies of determinant 1 preserving ${A}$, and ${SO_0(1,1)}$ to be the connected component of the identity. ${SO_0(1,1)}$ is then the group of matrices preserving both orientation and the sheets of the hyperbolas.

We can find an explicit form for the elements of ${SO_0(1,1)}$. Consider the matrix ${M = \left[ \begin{array}{cc} a & b \\ c& d \end{array} \right]}$. Writing down the equations ${M^TAM = A}$ and ${\det(M) = 1}$ gives us four equations, which we can solve to get the solutions

$\displaystyle \left[ \begin{array}{cc} \sqrt{b^2+1} & b \\ b & \sqrt{b^2+1} \end{array} \right] \textrm{ and } \left[ \begin{array}{cc} -\sqrt{b^2+1} & b \\ b & -\sqrt{b^2+1} \end{array} \right].$

Since we are interested in the connected component of the identity, we discard the solution on the right. It is useful to do a change of variables ${b = \sinh(t)}$, so we have (recall that ${\cosh^2(t) - \sinh^2(t) = 1}$).

$\displaystyle SO_0(1,1) = \left\{ \left[ \begin{array}{cc} \cosh(t) & \sinh(t) \\ \sinh(t) & \cosh(t) \end{array} \right] \, | \, t \in \mathbb{R} \right\}$

These matrices take ${\left[ \begin{array}{c} 0 \\ 1 \end{array} \right]}$ to ${\left[ \begin{array}{c} \sinh(t) \\ \cosh(t) \end{array} \right]}$. In other words, ${SO_0(1,1)}$ acts transitively on ${H}$ with trivial stabilizers, and in particular we have parmeterizing maps

$\displaystyle \mathbb{R} \rightarrow SO_0(1,1) \rightarrow H \textrm{ defined by } t \mapsto \left[ \begin{array}{cc} \cosh(t) & \sinh(t) \\ \sinh(t) & \cosh(t) \end{array} \right] \mapsto \left[ \begin{array}{c} \sinh(t) \\ \cosh(t) \end{array} \right]$

The first map is actually a Lie group isomorphism (with the group action on ${\mathbb{R}}$ being ${+}$) in addition to a diffeomorphism, since

$\displaystyle \left[ \begin{array}{cc} \cosh(t) & \sinh(t) \\ \sinh(t) & \cosh(t) \end{array} \right] \left[ \begin{array}{cc} \cosh(s) & \sinh(s) \\ \sinh(s) & \cosh(s) \end{array} \right] = \left[ \begin{array}{cc} \cosh(t+s) & \sinh(t+s) \\ \sinh(t+s) & \cosh(t+s) \end{array} \right]$

Metric

As mentioned above, ${\langle \cdot, \cdot \rangle_H}$ is not positive definite, but its restriction to the tangent space of ${H}$ is. We can see this in the following way: tangent vectors at a point ${p \in H}$ are characterized by the form ${\langle \cdot, \cdot \rangle_H}$. Specifically, ${v\in T_pH \Leftrightarrow \langle v, p \rangle_H}$, since (by a calculation) ${\frac{d}{dt} \langle p+tv, p+tv \rangle_H = 0 \Leftrightarrow \langle v, p \rangle_H}$. Therefore, ${SO_0(1,1)}$ takes tangent vectors to tangent vectors and preserves the form (and is transitive), so we only need to check that the form is positive definite on one tangent space. This is obvious on the tangent space to the point ${\left[ \begin{array}{c} 0 \\ 1 \end{array} \right]}$. Thus, ${H}$ is a Riemannian manifold, and ${SO_0(1,1)}$ acts by isometries.

Let’s use the parameterization ${\phi: t \mapsto \left[ \begin{array}{c} \sinh(t) \\ \cosh(t) \end{array} \right]}$. The unit (in the ${H}$ metric) tangent at ${\phi(t) = \left[ \begin{array}{c} \sinh(t) \\ \cosh(t) \end{array} \right]}$ is ${\left[ \begin{array}{c} \cosh(t) \\ \sinh(t) \end{array} \right]}$. The distance between the points ${\phi(s)}$ and ${\phi(t)}$ is

$\displaystyle d_H(\phi(s), \phi(t)) = \left| \int_s^t\sqrt{\langle \left[ \begin{array}{c} \cosh(t) \\ \sinh(t) \end{array} \right], \left[ \begin{array}{c} \cosh(t) \\ \sinh(t) \end{array} \right] \rangle_H dv } \right| = \left|\int_s^tdv \right| = |t-s|$

In other words, ${\phi}$ is an isometry from ${\mathbb{E}^1}$ to ${H}$.

1-dimensional hyperbollic space. The hyperboloid model is shown in blue, and the projective model is shown in red. An example of the projection map identifying ${H}$ with ${(-1,1) \subseteq \mathbb{R}\mathrm{P}^1}$ is shown.

2.2. Projective Model

Parameterizing

Real projective space ${\mathbb{R}\mathrm{P}^1}$ is the set of lines through the origin in ${\mathbb{R}^2}$. We can think about ${\mathbb{R}\mathrm{P}^1}$ as ${\mathbb{R} \cup \{\infty\}}$, where ${x\in \mathbb{R}}$ is associated with the line (point in ${\mathbb{R}\mathrm{P}^1}$) intersecting ${\{y=1\}}$ in ${x}$, and ${\infty}$ is the horizontal line. There is a natural projection ${\mathbb{R}^2 \setminus \{0\} \rightarrow \mathbb{R}\mathrm{P}^1}$ by projecting a point to the line it is on. Under this projection, ${H}$ maps to ${(-1,1)\subseteq \mathbb{R} \subseteq \mathbb{R}\mathrm{P}^1}$.

Since ${SO_0(1,1)}$ acts on ${\mathbb{R}^2}$ preserving the lines ${y = \pm x}$, it gives a projective action on ${\mathbb{R}\mathrm{P}^1}$ fixing the points ${\pm 1}$. Now suppose we have any projective linear isomorphism of ${\mathbb{R}\mathrm{P}^1}$ fixing ${\pm 1}$. The isomorphism is represented by a matrix ${A \in \mathrm{PGL}(2,\mathbb{R})}$ with eigenvectors ${\left[ \begin{array}{c} 1 \\ \pm 1 \end{array} \right]}$. Since scaling ${A}$ preserves its projective class, we may assume it has determinant 1. Its eigenvalues are thus ${\lambda}$ and ${\lambda^{-1}}$. The determinant equation, plus the fact that

$\displaystyle A \left[ \begin{array}{c} 1 \\ \pm 1 \end{array} \right] = \left[ \begin{array}{c} \lambda^{\pm 1} \\ \pm \lambda^{\pm 1} \end{array} \right]$

Implies that ${A}$ is of the form of a matrix in ${SO_0(1,1)}$. Therefore, the projective linear structure on ${(-1,1) \subseteq \mathbb{R}\mathrm{P}^1}$ is the “same” (has the same isometry (isomorphism) group) as the hyperbolic (Riemannian) structure on ${H}$.

Metric

Clearly, we’re going to use the pushforward metric under the projection of ${H}$ to ${(-1,1)}$, but it turns out that this metric is a natural choice for other reasons, and it has a nice expression.

The map taking ${H}$ to ${(-1,1) \subseteq \mathbb{R}\mathrm{P}^1}$ is ${\psi: \left[ \begin{array}{c} \sinh(t) \\ \cosh(t) \end{array} \right] \rightarrow \frac{\sinh(t)}{\cosh(T)} = \tanh(t)}$. The hyperbolic distance between ${x}$ and ${y}$ in ${(-1,1)}$ is then ${d_H(x,y) = |\tanh^{-1}(x) - \tanh^{-1}(y)|}$ (by the fact from the previous sections that ${\phi}$ is an isometry).

Recall the fact that ${\tanh(a\pm b) = \frac{\tanh(a) \pm \tanh(b)}{1 \pm \tanh(a)\tanh(b)}}$. Applying this, we get the nice form

$\displaystyle d_H(x,y) = \frac{y-x}{1 - xy}$

We also recall the cross ratio, for which we fix notation as ${ (z_1, z_2; z_3, z_4) := \frac{(z_3 -z_1)(z_4-z_2)}{(z_2-z_1)(z_4-z_3)}}$. Then

$\displaystyle (-1, x;y,1 ) = \frac{(y+1)(1-x)}{(x+1)(1-y)} = \frac{1-xy + (y-x)}{1-xy + (x-y)}$

Call the numerator of that fraction by ${N}$ and the denominator by ${D}$. Then, recalling that ${\tanh(u) = \frac{e^{2u}-1}{e^{2u}+1}}$, we have

$\displaystyle \tanh(\frac{1}{2} \log(-1,x;y,1)) = \frac{\frac{N}{D} -1}{\frac{N}{D} +1} = \frac{N-D}{N+D} = \frac{2(y-x)}{2(1-xy)} = \tanh(d_H(x,y))$

Therefore, ${d_H(x,y) = \frac{1}{2}\log(-1,x;y,-1)}$.

3. Hilbert Metric

Notice that the expression on the right above has nothing, a priori, to do with the hyperbolic projection. In fact, for any open convex body in ${\mathbb{R}\mathrm{P}^n}$, we can define the Hilbert metric on ${C}$ by setting ${d_H(p,q) = \frac{1}{2}\log(a,p,q,b)}$, where ${a}$ and ${b}$ are the intersections of the line through ${a}$ and ${b}$ with the boundary of ${C}$. How is it possible to take the cross ratio, since ${a,p,q,b}$ are not numbers? The line containing all of them is projectively isomorphic to ${\mathbb{R}\mathrm{P}^1}$, which we can parameterize as ${\mathbb{R} \cup \{\infty\}}$. The cross ratio does not depend on the choice of parameterization, so it is well defined. Note that the Hilbert metric is not necessarily a Riemannian metric, but it does make any open convex set into a metric space.

Therefore, we see that any open convex body in ${\mathbb{R}\mathrm{P}^n}$ has a natural metric, and the hyperbolic metric in ${H = (-1,1)}$ agrees with this metric when ${(-1,1)}$ is thought of as a open convex set in ${\mathbb{R}\mathrm{P}^1}$.

4. Higher-Dimensional Hyperbolic Space

4.1. Hyperboloid

The higher dimensional hyperbolic spaces are completely analogous to the 1-dimensional case. Consider ${\mathbb{R}^{n+1}}$ with the basis ${\{e_i\}_{i=1}^n \cup \{e\}}$ and the 2-form ${\langle v, w \rangle_H = \sum_{i=1}^n v_iw_i - v_{n+1}w_{n+1}}$. This is the form defined by the matrix ${J = I \oplus (-1)}$. Define ${\mathbb{H}^n}$ to be the positive (positive in the ${e}$ direction) sheet of the hyperbola ${\langle v,v\rangle_H = -1}$.

Let ${O(n,1)}$ be the linear transformations preserving the form, so ${O(n,1) = \{ A \, | \, A^TJA = J\}}$. This group is generated by ${O(1,1) \subseteq O(n,1)}$ as symmetries of the ${e_1, e}$ plane, together with ${O(n) \subseteq O(n,1)}$ as symmetries of the span of the ${e_i}$ (this subspace is euclidean). The group ${SO_0(n,1)}$ is the set of orientation preserving elements of ${O(n,1)}$ which preserve the positive sheet of the hyperboloid (${\mathbb{H}^n}$). This group acts transitively on ${\mathbb{H}^n}$ with point stabilizers ${SO(n)}$: this is easiest to see by considering the point ${(0,\cdots, 0, 1) \in \mathbb{H}^n}$. Here the stabilizer is clearly ${SO(n)}$, and because ${SO_0(n,1)}$ acts transitively, any stabilizer is a conjugate of this.

As in the 1-dimensional case, the metric on ${\mathbb{H}^n}$ is ${\langle \cdot , \cdot \rangle_H|_{T_p\mathbb{H}^n}}$, which is invariant under ${SO_0(n,1)}$.

Geodesics in ${\mathbb{H}^n}$ can be understood by consdering the fixed point sets of isometries, which are always totally geodesic. Here, reflection in a vertical (containing ${e}$) plane restricts to an (orientation-reversing, but that’s ok) isometry of ${\mathbb{H}^n}$, and the fixed point set is obviously the intersection of this plane with ${\mathbb{H}^n}$. Now ${SO_0(n,1)}$ is transitive on ${\mathbb{H}^n}$, and it sends planes to planes in ${\mathbb{R}^{n+1}}$, so we have a bijection

{Totally geodesic subspaces through ${p}$} ${\leftrightarrow}$ ${\mathbb{H}^n \cap}$ {linear subspaces of ${\mathbb{R}^{n+1}}$ through ${p}$ }

By considering planes through ${e}$, we can see that these totally geodesic subspaces are isometric to lower dimensional hyperbolic spaces.

4.2. Projective

Analogously, we define the projective model as follows: consider the disk ${\{v_{n+1} \,| v_{n+1} = 1, \langle v,v \rangle_H < 0\}}$. I.e. the points in the ${v_{n+1}}$ plane inside the cone ${\langle v,v \rangle_H = 0}$. We can think of ${\mathbb{R}\mathrm{P}^n}$ as ${\mathbb{R}^n \cup \mathbb{R}\mathrm{P}^{n-1}}$, so this disk is ${D^\circ \subseteq \mathbb{R}^n \subseteq \mathbb{R}\mathrm{P}^n}$. There is, as before, the natural projection of ${\mathbb{H}^n}$ to ${D^\circ}$, and the pushforward of the hyperbolic metric agrees with the Hilbert metric on ${D^\circ}$ as an open convex body in ${\mathbb{R}\mathrm{P}^n}$.

Geodesics in the projective model are the intersections of planes in ${\mathbb{R}^{n+1}}$ with ${D^\circ}$; that is, they are geodesics in the euclidean space spanned by the ${e_i}$. One interesting consequence of this is that any theorem which is true in euclidean geometry which does not reply on facts about angles is still true for hyperbolic space. For example, Pappus’ hexagon theorem, the proof of which does not use angles, is true.

4.3. Projective Model in Dimension 2

In the case that ${n=2}$, we can understand the projective isomorphisms of ${\mathbb{H}^2 = D \subseteq \mathbb{R}\mathrm{P}^2}$ by looking at their actions on the boundary ${\partial D}$. The set ${\partial D}$ is projectively isomorphic to ${\mathbb{R}\mathrm{P}^1}$ as an abstract manifold, but it should be noted that ${\partial D}$ is not a straight line in ${\mathbb{R}\mathrm{P}^2}$, which would be the most natural way to find ${\mathbb{R}\mathrm{P}^1}$‘s embedded in ${\mathbb{R}\mathrm{P}^2}$.

In addition, any projective isomorphism of ${\mathbb{R}\mathrm{P}^1 \cong \partial D}$ can be extended to a real projective isomorphism of ${\mathbb{R}\mathrm{P}^2}$. In other words, we can understand isometries of 2-dimensional hyperbolic space by looking at the action on the boundary. Since ${\partial D}$ is not a straight line, the extension is not trivial. We now show how to do this.

The automorphisms of ${\partial D \cong \mathbb{R}\mathrm{P}^1}$ are ${\mathrm{PSL}(2,\mathbb{R}}$. We will consider ${\mathrm{SL}(2,\mathbb{R})}$. For any Lie group ${G}$, there is an Adjoint action ${G \rightarrow \mathrm{Aut}(T_eG)}$ defined by (the derivative of) conjugation. We can similarly define an adjoint action ${\mathrm{ad}}$ by the Lie algebra on itself, as ${\mathrm{ad}(\gamma '(0)) := \left. \frac{d}{dt} \right|_{t=0} \mathrm{Ad}(\gamma(t))}$ for any path ${\gamma}$ with ${\gamma(0) = e}$. If the tangent vectors ${v}$ and ${w}$ are matrices, then ${\mathrm{ad}(v)(w) = [v,w] = vw-wv}$.

We can define the Killing form ${B}$ on the Lie algebra by ${B(v,w) = \mathrm{Tr}(\mathrm{ad}(v)\mathrm{ad}(w))}$. Note that ${\mathrm{ad}(v)}$ is a matrix, so this makes sense, and the Lie group acts on the tangent space (Lie algebra) preserving this form.

Now let’s look at ${\mathrm{SL}(2,\mathbb{R})}$ specifically. A basis for the tangent space (Lie algebra) is ${e_1 = \left[ \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right]}$, ${e_2 = \left[ \begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array} \right]}$, and ${e_3 = \left[ \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right]}$. We can check that ${[e_1,e_2] = e_3}$, ${[e_1,e_3] = -2e_1}$, and ${[e_2, e_3]=2e_2}$. Using these relations plus the antisymmetry of the Lie bracket, we know

$\displaystyle \mathrm{ad}(e_1) = \left[ \begin{array}{ccc} 0 & 0 & -2 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right] \qquad \mathrm{ad}(e_2) = \left[ \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 2 \\ -1 & 0 & 0 \end{array}\right] \qquad \mathrm{ad}(e_3) = \left[ \begin{array}{ccc} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 0 \end{array}\right]$

Therefore, the matrix for the Killing form in this basis is

$\displaystyle B_{ij} = B(e_i,e_j) = \mathrm{Tr}(\mathrm{ad}(e_i)\mathrm{ad}(e_j)) = \left[ \begin{array}{ccc} 0 & 4 & 0 \\ 4 & 0 & 0 \\ 0 & 0 & 8 \end{array}\right]$

This matrix has 2 positive eigenvalues and one negative eigenvalue, so its signature is ${(2,1)}$. Since ${\mathrm{SL}(2,\mathbb{R})}$ acts on ${T_e(\mathrm{SL}(2,\mathbb{R}))}$ preserving this form, we have ${\mathrm{SL}(2,\mathbb{R}) \cong O(2,1)}$, otherwise known at the group of isometries of the disk in projective space ${\mathbb{R}\mathrm{P}^2}$, otherwise known as ${\mathbb{H}^2}$.

Any element of ${\mathrm{PSL}(2,\mathbb{R})}$ (which, recall, was acting on the boundary of projective hyperbolic space ${\partial D}$) therefore extends to an element of ${O(2,1)}$, the isometries of hyperbolic space, i.e. we can extend the action over the disk.

This means that we can classify isometries of 2-dimensional hyperbolic space by what they do to the boundary, which is determined generally by their eigevectors (${\mathrm{PSL}(2,\mathbb{R})}$ acts on ${\mathbb{R}\mathrm{P}^1}$ by projecting the action on ${\mathbb{R}^2}$, so an eigenvector of a matrix corresponds to a fixed line in ${\mathbb{R}^2}$, so a fixed point in ${\mathbb{R}\mathrm{P}^1 \cong \partial D}$. For a matrix ${A}$, we have the following:

• ${|\mathrm{Tr}(A)| < 2}$ (elliptic) In this case, there are no real eigenvalues, so no real eigenvectors. The action here is rotation, which extends to a rotation of the entire disk.
• ${|\mathrm{Tr}(A)| = 2}$ (parabolic) There is a single real eigenvector. There is a single fixed point, to which all other points are attracted (in one direction) and repelled from (in the other). For example, the action in projective coordinates sending ${[x:y]}$ to ${[x+1:y]}$: infinity is such a fixed point.
• ${|\mathrm{Tr}(A)| > 2}$ (hyperbolic) There are two fixed point, one attracting and one repelling.
•

5. Complex Hyperbolic Space

We can do a construction analogous to real hyperbolic space over the complexes. Define a Hermitian form ${q}$ on ${\mathbb{C}^{n+1}}$ with coordinates ${\{z_1,\cdots, z_n\} \cup \{w\}}$ by ${q(x_1,\cdots x_n, w) = |z_1|^2 + \cdots + |z_n|^2 - |w|^2}$. We will also refer to ${q}$ as ${\langle \cdot, \cdot \rangle_q}$. The (complex) matrix for this form is ${J = I \oplus (-1)}$, where ${q(v,w) = v^*Jw}$. Complex linear isomorphisms preserving this form are matrices ${A}$ such that ${A^*JA = J}$. This is our definition for ${\mathrm{U}(q) := \mathrm{U}(n,1)}$, and we define ${\mathrm{SU}(n,1)}$ to be those elements of ${\mathrm{U}(n,1)}$ with determinant of norm 1.

The set of points ${z}$ such that ${q(z) = -1}$ is not quite what we are looking for: first it is a ${2n+1}$ real dimensional manifold (not ${2n}$ as we would like for whatever our definition of “complex hyperbolic ${n}$ space” is), but more importantly, ${q}$ does not restrict to a positive definite form on the tangent spaces. Call the set of points ${z}$ where ${q(z) = -1}$ by ${\bar{H}}$. Consider a point ${p}$ in ${\bar{H}}$ and ${v}$ in ${T_p\bar{H}}$. As with the real case, by the fact that ${v}$ is in the tangent space,

$\displaystyle \left. \frac{d}{dt} \right|_{t=0} \langle p + tv, p+tv\rangle_q = 0 \quad \Rightarrow \quad \langle v, p \rangle_q + \langle p,v \rangle_q = 0$

Because ${q}$ is hermitian, the expression on the right does not mean that ${\langle v,p\rangle_q = 0}$, but it does mean that ${\langle v,p \rangle_q}$ is purely imaginary. If ${\langle v,p \rangle_q = ik}$, then ${\langle v,v\rangle_q < 0}$, i.e. ${q}$ is not positive definite on the tangent spaces.

However, we can get rid of this negative definite subspace. ${S^1}$ as the complex numbers of unit length (or ${\mathrm{U}(1)}$, say) acts on ${\mathbb{C}^{n+1}}$ by multiplying coordinates, and this action preserves ${q}$: any phase goes away when we apply the absolute value. The quotient of ${\bar{H}}$ by this action is ${\mathbb{C}\mathbb{H}^n}$. The isometry group of this space is still ${\mathrm{U}(n,1)}$, but now there are point stabilizers because of the action of ${\mathrm{U}(1)}$. We can think of ${\mathrm{U}(1)}$ inside ${\mathrm{U}(n,1)}$ as the diagonal matrices, so we can write

$\displaystyle \mathrm{SU}(n,1) \times \mathrm{U}(1) \cong U(n,1)$

And the projectivized matrices ${\mathrm{PSU}(n,1)}$ is the group of isometries of ${\mathbb{C}\mathbb{H}^n \subseteq \mathbb{C}^n \subseteq \mathbb{C}\mathrm{P}^n}$, where the middle ${\mathbb{C}^n}$ is all vectors in ${\mathbb{C}^{n+1}}$ with ${w=1}$ (which we think of as part of complex projective space). We can also approach this group by projectivizing, since that will get rid of the unwanted point stabilizers too: we have ${\mathrm{PU}(n,1) \cong \mathrm{PSU}(n,1)}$.

5.1. Case ${n=1}$

In the case ${n=1}$, we can actually picture ${\mathbb{C}\mathrm{P}^1}$. We can’t picture the original ${\mathbb{C}^4}$, but we are looking at the set of ${(z,w)}$ such that ${|z|^2 - |w|^2 = -1}$. Notice that ${|w| \ge 1}$. After projectivizing, we may divide by ${w}$, so ${|z/w| - 1 = -1/|w|}$. The set of points ${z/w}$ which satisfy this is the interior of the unit circle, so this is what we think of for ${\mathbb{C}\mathbb{H}^1}$. The group of complex projective isometries of the disk is ${\mathrm{PU}(1,1)}$. The straight horizontal line is a geodesic, and the complex isometries send circles to circles, so the geodesics in ${\mathbb{C}\mathbb{H}^1}$ are circles perpendicular to the boundary of ${S^1}$ in ${\mathbb{C}}$.

Imagine the real projective model as a disk sitting at height one, and the geodesics are the intersections of planes with the disk. Complex hyperbolic space is the upper hemisphere of a sphere of radius one with equator the boundary of real hyperbolic space. To get the geodesics in complex hyperbolic space, intersect a plane with this upper hemisphere and stereographically project it flat. This gives the familiar Poincare disk model.

5.2. Real ${\mathbb{H}^2}$‘s contained in ${\mathbb{C}\mathbb{H}^n}$

${\mathbb{C}\mathbb{H}^2}$ contains 2 kinds of real hyperbolic spaces. The subset of real points in ${\mathbb{C}\mathbb{H}^n}$ is (real) ${\mathbb{H}^n}$, so we have a many ${\mathbb{H}^2 \subseteq \mathbb{H}^n \subseteq \mathbb{C}\mathbb{H}^n}$. In addition, we have copies of ${\mathbb{C}\mathbb{H}^1}$, which, as discussed above, has the same geometry (i.e. has the same isometry group) as real ${\mathbb{H}^2}$. However, these two real hyperbolic spaces are not isometric. the complex hyperbolic space ${\mathbb{C}\mathbb{H}^1}$ has a more negative curvature than the real hyperbolic spaces. If we scale the metric on ${\mathbb{C}\mathbb{H}^n}$ so that the real hyperbolic spaces have curvature ${-1}$, then the copies of ${\mathbb{C}\mathbb{H}^1}$ will have curvature ${-4}$.

In a similar vein, there is a symplectic structure on ${\mathbb{C}\mathbb{H}^n}$ such that the real ${\mathbb{H}^2}$ are lagrangian subspaces (the flattest), and the ${\mathbb{C}\mathbb{H}^1}$ are symplectic, the most negatively curved.

An important thing to mention is that complex hyperbolic space does not have constant curvature(!).

6. Poincare Disk Model and Upper Half Space Model

The projective models that we have been dealing with have many nice properties, especially the fact that geodesics in hyperbolic space are straight lines in projective space. However, the angles are wrong. There are models in which the straight lines are “curved” i.e. curved in the euclidean metric, but the angles between them are accurate. Here we are interested in a group of isometries which preserves angles, so we are looking at a conformal model. Dimension 2 is special, because complex geometry is real conformal geometry, but nevertheless, there is a model of ${\mathbb{R}\mathbb{H}^n}$ in which the isometries of the space are conformal.

Consider the unit disk ${D^n}$ in ${n}$ dimensions. The conformal automorphisms are the maps taking (straight) diameters and arcs of circles perpendicular to the boundary to this same set. This model is abstractly isomorphic to the Klein model in projective space. Imagine the unit disk in a flat plane of height one with an upper hemisphere over it. The geodesics in the Klein model are the intersections of this flat plane with subspaces (so they are straight lines, for example, in dimension 2). Intersecting vertical planes with the upper hemisphere and stereographically projecting it flat give geodesics in the Poincare disk model. The fact that this model is the “same” (up to scaling the metric) as the example above of ${\mathbb{C}\mathbb{H}^1}$ is a (nice) coincidence.

The Klein model is the flat disk inside the sphere, and the Poincare disk model is the sphere. Geodesics in the Klein model are intersections of subspaces (the angled plane) with the flat plane at height 1. Geodesics in the Poincare model are intersections of vertical planes with the upper hemisphere. The two darkened geodesics, one in the Klein model and one in the Poincare, correspond under orthogonal projection. We get the usual Poincare disk model by stereographically projecting the upper hemisphere to the disk. The projection of the geodesic is shown as the curved line inside the disk

The Poincare disk model. A few geodesics are shown.

Now we have the Poincare disk model, where the geodesics are straight diameters and arcs of circles perpendicular to the boundary and the isometries are the conformal automorphisms of the unit disk. There is a conformal map from the disk to an open half space (we typically choose to conformally identify it with the upper half space). Conveniently, the hyperbolic metric on the upper half space ${d_H}$ can be expressed at a point ${(x,t)}$ (euclidean coordinates) as ${d_H = d_E/t}$. I.e. the hyperbolic metric is just a rescaling (at each point) of the euclidean metric.

One of the important things that we wanted in our models was the ability to realize isometries of the model with isometries of the ambient space. In the case of a one-parameter family of isometries of hyperbolic space, this is possible. Suppose that we have a set of elliptic isometries. Then in the disk model, we can move that point to the origin and realize the isometries by rotations. In the upper half space model, we can move the point to infinity, and realize them by translations.

I am (update: was) currently (update: but am no longer) in Brisbane for the “New directions in geometric group theory” conference, which has been an entirely enjoyable and educational experience. I got to eat fish and chips, to watch Australia make 520 for 7 (declared) against the West Indies at the WACA, and to hear Masato Mimura give a very nice talk about his recent results on rigidity of the “universal lattice”.

His talk included a quick and beautiful survey of some geometric aspects of the theory of rigidity for infinite groups, which I will attempt to partially reproduce (despite the limitations of the wordpress format). In this context, rigidity is expressed in terms of isometric affine actions of groups on Banach spaces. This means the following. Suppose $B$ is a Banach space (i.e. a complete, normed vector space) and $G$ is a group. A linear isometric action is a representation $\rho$ from $G$ to the group of linear isometries of $B$ — i.e. linear norm-preserving automorphisms. An affine action is a representation from $G$ to the group of affine isometries of $B$ — i.e. isometries as a metric space that do not necessarily fix the zero element. The group of isometries of a Banach space $B$ is a semi-direct product $B \rtimes U(B)$ where $U(B)$ is the group of linear isometries, and $B$ is the Banach space, thought of as an Abelian group, acting on itself by (isometric) translations. Such an action is usually encoded by a pair $\rho:G \to U(B)$ which records the “linear” part of the action, and a 1-cocycle with coefficients in $\rho$, i.e. a function $c:G \to B$ satisfying $c(gh) = c(g) + \rho(g)c(h)$ for every $g,h \in G$. This formula might look strange if you don’t know where it comes from: it is just the way that factors transform in semi-direct products. The affine action is given by sending $g \in G$ to the transformation that sends each $b \in B$ to $\rho(g)b + c(g)$. Consequently, $gh$ is sent to the transformation that sends $b$ to $\rho(gh)b + c(gh)$ and the fact that this is a group action becomes the formula

$\rho(gh)b + c(gh) = \rho(g)(\rho(h)b + c(h)) + c(g) = \rho(gh)b + \rho(g)c(h) + c(g)$

Equating the left and right hand sides gives the cocycle condition. Given one affine isometric action, one can obtain another in a silly way by conjugating by an isometry $b \to b + b'$ for some $b' \in B$. Under conjugation by such an isometry, a cocycle $c$ transforms by $c(g) \to c(g) + \rho(g)b' - b'$. A function of the form $c(g) = \rho(g)b' - b'$ is called a 1-coboundary, and the quotient of the space of 1-cocycles by the space of 1-coboundaries is the 1 dimensional cohomology of $G$ with coefficients in $\rho:G \to U(B)$. This is usually denoted $H^1(G,\rho)$, where $B$ is suppressed in the notation. In particular, an affine isometric action of $G$ on $B$ with linear part $\rho$ has a global fixed point iff it represents $0$ in $H^1(G,\rho)$. Contrapositively, $G$ admits an affine isometric action on $B$ without a global fixed point iff $H^1(G,\rho) \ne 0$ for some $\rho$.

A group $G$ is said to satisfy Serre’s Property (FH) if every affine isometric action of $G$ on a Hilbert space has a global fixed point. In 2007, Bader-Furman-Gelander-Monod introduced a property (FB) for a group $G$ to mean that every affine isometric action of $G$ on some (out of a class of) Banach space(s) $B$ has a global fixed point. Mimura used the notation property (FL_p) for the case that $B$ is allowed to range over the class of $L_p$ spaces (for some fixed $1 < p < \infty$).

Intimately related is Kazhdan’s Property (T), introduced by Kazhdan in this paper. Let $G$ be a locally compact topological group (for example, a discrete group). The set of irreducible unitary representations of $G$ is called its dual, and denoted $\hat{G}$. This dual is topologized in the following way. Associated to a representation $\rho:G \to U(L)$, a unit vector $X \in L$, a positive number $\epsilon > 0$ and a compact subset $K \subset G$ there is an open neighborhood of $\rho$ consisting of representations $\rho':G \to U(L')$ for which there is a unit vector $Y \in L$ such that $|\langle \rho(g)X,X\rangle - \langle \rho(g')Y, Y\rangle| < \epsilon$ whenever $g \in K$. With this topology (called the Fell topology), one says that a group $G$ has property (T) if the trivial representation is isolated in $\hat{G}$. Note that this topology is very far from being Hausdorff: the trivial representation fails to be isolated exactly when there are a sequence of representations $\rho_i:G \to U(L_i)$, unit vectors $X_i \in L_i$, numbers $\epsilon_i \to 0$ and compact sets $K_i$ exhausting $G$ so that $|\langle\rho_i(g)X_i,X_i\rangle| < \epsilon_i$ for any $g \in K_i$. The vectors $X_i$ are said to be (a sequence of) almost invariant vectors. Hence (informally) a group has property (T) if some compact subset must move some unit vector a definite amount in every irreducible nontrivial unitary representation. If a group fails to have property (T), one can rescale a sequence of irreducible actions near a sequence of almost invariant vectors in such a way that one obtains in the geometric limit a nontrivial isometric action on $L^2$ without a global fixed point. A famous theorem of Delorme-Guichardet says that property (T) and property (FH) are equivalent for (locally compact second countable) groups. Property (T) passes to quotients, and to lattices (i.e. finite covolume discrete subgroups of a topological group). Kazhdan already showed in his paper that $\text{SL}(n,\mathbb{R})$ has property (T) for $n$ at least $3$, and therefore the same is true for lattices in this groups, such as $\text{SL}(n,\mathbb{Z})$, a fact which is not easy to see directly from the definition. One beautiful application, already pointed out by Kazhdan, is that this means that all lattices in $\text{SL}(n,\mathbb{R})$, for instance the groups $\text{SL}(n,\mathbb{Z})$ (and in fact, all discrete groups with property (T)) are finitely generated. Kazhdan’s proof of this is incredibly short: let $G$ be a discrete group and $g_i$ and sequence of elements. For each $i$, let $G_i$ be the subgroup of $G$ generated by $\lbrace g_1,g_2,\cdots,g_i\rbrace$. Notice that $G$ is finitely generated iff $G_i=G$ for all sufficiently large $i$. On the other hand, consider the unitary representations of $G$ induced by the trivial representations on the $G_i$. Every compact subset of $G$ is finite, and therefore eventually fixes a vector in every one of these representations; thus there is a sequence of almost fixed vectors. If $G$ has property (T), this sequence eventually contains a fixed vector, which can only happen if $G/G_i$ is finite, in which case $G$ is finitely generated, as claimed.

Property (FL_p) generalizes (FH) (equivalently (T)) in many significant ways, with interesting applications to dynamics. For example, Navas showed that if $G$ is a group with property (T) then every action of $G$ on a circle which is at least $C^{1+1/2 + \epsilon}$ factors through a finite group. Navas’s argument can be generalized straightforwardly to show that if $G$ has (FL_p) for some $p>2$ then every action of $G$ on a circle which is at least $C^{1+1/p+\epsilon}$ factors through a finite group. The proof rests on a beautiful construction due to Reznikov (although a similar construction can be found in Pressley-Segal) of certain functions on a configuration space of the circle which are not in $L^p$ but have coboundaries which are; this gives rise to nontrivial cohomology with $L^p$ coefficients for groups acting on the circle in a sufficiently interesting way.

(Update: Nicolas Monod points out in an email that the “function on a configuration space” is morally just the derivative. In fact, he made the nice remark that if $D$ is any elliptic operator on an $n$-manifold, then the commutator $[D,g]$ is of Schatten class $(n+1)$ whenever $g$ is a sufficiently smooth function; morally this should give rise to nontrivial cohomology with suitable coefficients for groups acting with enough regularity on any given $n$-manifold, and one would like to use this e.g. to approach Zimmer’s conjecture, but nobody seems to know how to make this work as yet; in fact the work of Monod et. al. on (FL_p) is at least partly motivated by this general picture.)

Mimura discussed a spectrum of rigid behaviour for infinite groups, ranging from most rigid (property (FL_p) for every $p$) to least rigid (amenable) (note: every finite group is both amenable and has property (T), so this only really makes sense for infinite groups; moreover, every reasonable measure of rigidity for infinite groups is usually invariant under passing to subgroups of finite index). Free groups, $\text{SL}(2,\mathbb{Z})$ and so on are very non-rigid. However, it is well-known that certain infinite families of (word) hyperbolic groups, including lattices in groups of isometries of quaternion-hyperbolic symmetric spaces, and “random” groups with relations having density parameter $1/3 < d < 1/2$ (see Zuk or Ollivier) are both hyperbolic and have property (T). Nevertheless, these groups are not as rigid as higher rank lattices like $\text{SL}(n,\mathbb{Z})$ for $n>2$. The latter have property (FL_p) for every $1< p < \infty$, whereas Yu showed that every hyperbolic group admits a proper affine isometric action on $\ell^p$ for some $p$ (the existence of a proper affine isometric action on a Hilbert space is called “a-T-menability” by Gromov, and the “Haagerup property” by some. Groups satisfying this property, or even Yu’s weaker property, are known to satisfy some version of the Baum-Connes conjecture, the subject of a very nice minicourse by Graham Niblo at the same conference).

It is in this context that one can appreciate Mimura’s results. His first main result is that the group $\text{SL}_n(\mathbb{Z}[x_1,x_2,\cdots,x_n])$ (i.e. the “universal lattice”) has property (FL_p) for every $1 provided $n$ is at least 4. Since property (FL_p) (like (T)) passes to quotients, this implies that $\text{SL}_n(R)$ has (FL_p) for every unital, commutative, finitely generated ring $R$.

His second main result concerns a “quasification” of FL_p, to a property called (FFL_p). Without getting too technical, this property concerns “quasi-actions” of a group on a Banach space by affine isometries; algebraically these are encoded by 1-cochains $c:G \to B$ for which there is a universal constant $D$ so that $|c(gh) - c(g) -\rho(g)c(h)| < D$ as measured in the Banach norm on $B$. Any bounded map $c:G \to B$ defines a 1-cochain; such (bounded) 1-cochains corresponds to  quasi-action with a bounded orbit. Associated to $\rho: G \to U(B)$ one defines in a similar way a complex of bounded cochains; quasi-actions modulo bounded quasi-actions are parameterized by the kernel of the comparison map $H^2_b(G,\rho) \to H^2(G,\rho)$ from bounded to ordinary cohomology. Mimura’s second main result is that when $G$ is the universal lattice as above, and $\rho$ has no invariant vectors, the comparison map from bounded to ordinary cohomology in dimension 2 is injective.

The fact that $\rho$ as above is required to have no invariant vectors is a technical necessity of Mimura’s proof. When $\rho$ is trivial, one is studying “ordinary” bounded cohomology, and there is an exact sequence

$0 \to H^1(G) \to Q(G) \to H^2_b(G) \to H^2(G)$

with real coefficients for any $G$ (here $Q(G)$ denotes the vector space of homogeneous quasimorphisms on $G$). In this context, one knows by Bavard duality that $H^2_b \to H^2$ is injective if and only if the stable commutator length is identically zero on $[G,G]$. By quite a different method, Mimura shows that for $n$ at least $6$, and for any Euclidean ring $R$ (i.e. a ring for which one has a Euclidean algorithm; for example, $R = \mathbb{C}[x]$) the group $SL_n(R)$ has vanishing stable commutator length, and therefore one has injectivity of bounded to ordinary cohomology in dimension $2$.

(Update 1/9/2010): Nicholas Monod sent me a nice email commenting on a couple of points in this blog entry, and I have consequently modified the language a bit in a few places. Ta much!

On page 10 of Besse’s famous book on Einstein manifolds one finds the following quote:

It would seem that Riemannian and Lorentzian geometry have much in common: canonical connections, geodesics, curvature tensor, etc. . . . But in fact this common part is only a common disposition at the onset: one soon enters different realms.

I will not dispute this. But it is not clear to me whether this divergence is a necessary consequence of the nature of the objects of study (in either case), or an artefact of the schism between mathematics and physics during much of the 20th century. In any case, in this blog post I have the narrow aim of describing some points of contact between Lorentzian (and more generally, causal) geometry and other geometries (hyperbolic, symplectic), which plays a significant role in some of my research.

The first point of contact is the well-known duality between geodesics in the hyperbolic plane and points in the (projectivized) “anti de-Sitter plane”. Let $\mathbb{R}^{2,1}$ denote a $3$-dimensional vector space equipped with a quadratic form

$q(x,y,z) = x^2 + y^2 - z^2$

If we think of the set of rays through the origin as a copy of the real projective plane $\mathbb{RP}^2$, the hyperbolic plane is the set of projective classes of vectors $v$ with $q(v)<0$, the (projectivized) anti de-Sitter plane is the set of projective classes of vectors $v$ with $q(v)>0$, and their common boundary is the set of projective classes of (nonzero) vectors $v$ with $q(v)=0$. Topologically, the hyperbolic plane is an open disk, the anti de-Sitter plane is an open Möbius band, and their boundary is the “ideal circle” (note: what people usually call the anti de-Sitter plane is actually the annulus double-covering this Möbius band; this is like the distinction between spherical geometry and elliptic geometry). Geometrically, the hyperbolic plane is a complete Riemannian surface of constant curvature $-1$, whereas the anti de-Sitter plane is a complete Lorentzian surface of constant curvature $-1$.

In this projective model, a hyperbolic geodesic $\gamma$ is an open straight line segment which is compactified by adding an unordered pair of points in the ideal circle. The straight lines in the anti de-Sitter plane tangent to the ideal circle at these two points intersect at a point $p_\gamma$. Moreover, the set of geodesics $\gamma$ in the hyperbolic plane passing through a point $q$ are dual to the set of points $p_\gamma$ in the anti de-Sitter plane that lie on a line which does not intersect the ideal circle. In the figure, three concurrent hyperbolic geodesics are dual to three colinear anti de-Sitter points.

The anti de-Sitter geometry has a natural causal structure. There is a cone field whose extremal vectors at every point $p$ are tangent to the straight lines through $p$ that are also tangent to the ideal circle. A smooth curve is timelike if its tangent at every point is supported by this cone field, and spacelike if its tangent is everywhere not supported by the cone field. A timelike curve corresponds to a family of hyperbolic geodesics which locally intersect each other; a spacelike curve corresponds to a family of disjoint hyperbolic geodesics that foliate some region.

One can distinguish (locally) between future and past along a timelike trajectory, by (arbitrarily) identifying the “future” direction with a curve which winds positively around the ideal circle. The fact that one can distinguish in a consistent way between the positive and negative direction is equivalent to the existence of a nonzero section of timelike vectors. On the other hand, there does not exist a nonzero section of spacelike vectors, so one cannot distinguish in a consistent way between left and right (this is a manifestation of the non-orientability of the Möbius band).

The duality between the hyperbolic plane and the anti de-Sitter plane is a manifestation of the fact that (at least at the level of Lie algebras) they have the same (infinitesimal) symmetries. Let $O(2,1)$ denote the group of real $3\times 3$ matrices which preserve $q$; i.e. matrices $A$ for which $q(A(v)) = q(v)$ for all vectors $v$. This contains a subgroup $SO^+(2,1)$ of index $4$ which preserves the “positive sheet” of the hyperboloid $q=-1$, and acts on it in an orientation-preserving way. The hyperbolic plane is the homogeneous space for this group whose point stabilizers are a copy of $SO(2)$ (which acts as an elliptic “rotation” of the tangent space to their common fixed point). The anti de-Sitter plane is the homogeneous space for this group whose point stabilizers are a copy of $SO^+(1,1)$ (which acts as a hyperbolic “translation” of the geodesic in hyperbolic space dual to the given point in anti de-Sitter space). The ideal circle is the homogeneous space whose point stabilizers are a copy of the affine group of the line. The hyperbolic plane admits a natural Riemannian metric, and the anti de-Sitter plane a Lorentz metric, which are invariant under these group actions. The causal structure on the anti de-Sitter plane limits to a causal structure on the ideal circle.

Now consider the $4$-dimensional vector space $\mathbb{R}^{2,2}$ and the quadratic form $q(v) = x^2 + y^2 - z^2 - w^2$. The ($3$-dimensional) sheets $q=1$ and $q=-1$ both admit homogeneous Lorentz metrics whose point stabilizers are copies of $SO^+(1,2)$ and $SO^+(2,1)$ (which are isomorphic but sit in $SO(2,2)$ in different ways). These $3$-manifolds are compactified by adding the projectivization of the cone $q=0$. Topologically, this is a Clifford torus in $\mathbb{RP}^3$ dividing this space into two open solid tori which can be thought of as two Lorentz $3$-manifolds. The causal structure on the pair of Lorentz manifolds limits to a pair of complementary causal structures on the Clifford torus. (edited 12/10)

Let’s go one dimension higher, to the $5$-dimensional vector space $\mathbb{R}^{2,3}$ and the quadratic form $q(v) = x^2 + y^2 - u^2 - z^2 - w^2$. Now only the sheet $q=1$ is a Lorentz manifold, whose point stabilizers are copies of $SO^+(1,3)$, with an associated causal structure. The projectivized cone $q=0$ is a non-orientable twisted $S^2$ bundle over the circle, and it inherits a causal structure in which the sphere factors are spacelike, and the circle direction is timelike. This ideal boundary can be thought of in quite a different way, because of the exceptional isomorphism at the level of (real) Lie algebras $so(2,3)= sp(4)$, where $sp(4)$ denotes the Lie algebra of the symplectic group in dimension $4$. In this manifestation, the ideal boundary is usually denoted $\mathcal{L}_2$, and can be thought of as the space of Lagrangian planes in $\mathbb{R}^4$ with its usual symplectic form. One way to see this is as follows. The wedge product is a symmetric bilinear form on $\Lambda^2 \mathbb{R}^4$ with values in $\Lambda^4 \mathbb{R}^4 = \mathbb{R}$. The associated quadratic form vanishes precisely on the “pure” $2$-forms — i.e. those associated to planes. The condition that the wedge of a given $2$-form with the symplectic form vanishes imposes a further linear condition. So the space of Lagrangian $2$-planes is a quadric in $\mathbb{RP}^4$, and one may verify that the signature of the underlying quadratic form is $(2,3)$. The causal structure manifests in symplectic geometry in the following way. A choice of a Lagrangian plane $\pi$ lets us identify symplectic $\mathbb{R}^4$ with the cotangent bundle $T^*\pi$. To each symmetric homogeneous quadratic form $q$ on $\pi$ (thought of as a smooth function) is associated a linear Lagrangian subspace of $T^*\pi$, namely the (linear) section $dq$. Every Lagrangian subspace transverse to the fiber over $0$ is of this form, so this gives a parameterization of an open, dense subset of $\mathcal{L}_2$ containing the point $\pi$. The set of positive definite quadratic forms is tangent to an open cone in $T_\pi \mathcal{L}_2$; the field of such cones as $\pi$ varies defines a causal structure on $\mathcal{L}_2$ which agrees with the causal structure defined above.

These examples can be generalized to higher dimension, via the orthogonal groups $SO(n,2)$ or the symplectic groups $Sp(2n,\mathbb{R})$. As well as two other infinite families (which I will not discuss) there is a beautiful “sporadic” example, connected to what Freudenthal called octonion symplectic geometry associated to the noncompact real form $E_7(-25)$ of the exceptional Lie group, where the ideal boundary $S^1\times E_6/F_4$ has an invariant causal structure whose timelike curves wind around the $S^1$ factor; see e.g. Clerc-Neeb for a more thorough discussion of the theory of Shilov boundaries from the causal geometry point of view, or see here or here for a discussion of the relationship between the octonions and the exceptional Lie groups.

The causal structure on these ideal boundaries gives rise to certain natural $2$-cocycles on their groups of automorphisms. Note in each case that the ideal boundary has the topological structure of a bundle over $S^1$ with spacelike fibers. Thus each closed timelike curve has a well-defined winding number, which is just the number of times it intersects any one of these spacelike slices. Let $C$ be an ideal boundary as above, and let $\tilde{C}$ denote the cyclic cover dual to a spacelike slice. If $p$ is a point in $\tilde{C}$, we let $p+n$ denote the image of $p$ under the $n$th power of the generator of the deck group of the covering. If $g$ is a homeomorphism of $C$ preserving the causal structure, we can lift $g$ to a homeomorphism $\tilde{g}$ of $\tilde{C}$. For any such lift, define the rotation number of $\tilde{g}$ as follows: for any point $p \in \tilde{C}$ and any integer $n$, let $r_n$ be the the smallest integer for which there is a causal curve from $p$ to $\tilde{g}(p)$ to $p+r_n$, and then define $rot(\tilde{g}) = \lim_{n \to \infty} r_n/n$. This function is a quasimorphism on the group of causal automorphisms of $\tilde{C}$, with defect equal to the least integer $n$ such that any two points $p,q$ in $C$ are contained in a closed causal loop with winding number $n$. In the case of the symplectic group $Sp(2n,\mathbb{R})$ with causal boundary $\mathcal{L}_n$, the defect is $n$, and the rotation number is (sometimes) called the symplectic rotation number; it is a quasimorphism on the universal central extension of $Sp(2n,\mathbb{R})$, whose coboundary descends to the Maslov class (an element of $2$-dimensional bounded cohomology) on the symplectic group.

Causal structures in groups of symplectomorphisms or contactomorphisms are intensely studied; see for instance this paper by Eliashberg-Polterovich.

A geometric structure on a manifold is an atlas of charts with values in some kind of “model space”, and transformation functions taken from some pseudogroup of transformations on the model space. If $X$ is the model space, and $G$ is the pseudo-group, one talks about a $(G,X)$-structure on a manifold $M$. One usually (but not always) wants $X$ to be homogeneous with respect to $G$. So, for instance, one talks about smooth structures, conformal structures, projective structures, bilipschitz structures, piecewise linear structures, symplectic structures, and so on, and so on. Riemannian geometry does not easily fit into this picture, because there are so few (germs of) isometries of a typical Riemannian metric, and so many local invariants; but Riemannian metrics modeled on a locally symmetric space, with $G$ a Lie group of symmetries of $X$, are a very significant example.

Sometimes the abstract details of a theory are hard to grasp before looking at some fundamental examples. The case of geometric structures on $1$-manifolds is a nice example, which is surprisingly rich in some ways.

One of the most important ways in which geometric structures arise is in the theory of ODE’s. Consider a first order ODE in one variable, e.g. an equation like $y' = f(y,t)$. If we fix an “initial” value $y(t_0)=y_0$, then we are guaranteed short time existence and uniqueness of a solution (providing the function $f$ is nice enough). But if we do not fix an initial value, we can instead think of an ODE as a $1$-parameter family of (perhaps partially defined) maps from $\mathbb{R}$ to itself. For each fixed $t$, the function $f(y,t)$ defines a vector field on $\mathbb{R}$. We can think of the ODE as specifying a path in the Lie algebra of vector fields on $\mathbb{R}$; solving the ODE amounts to finding a path in the Lie group of diffeomorphisms of $\mathbb{R}$ (or some partially defined Lie pseudogroup of diffeomorphisms on some restricted subdomain) which is tangent to the given family of vector fields. It makes sense therefore to study special classes of equations, and ask when this family of maps is conjugate into an interesting pseudogroup; equivalently, that the evolution of the solutions preserves an interesting geometric structure on $\mathbb{R}$. We consider some examples in turn.

1. Indefinite integral $y' = a(t)$. The group in this case is $\mathbb{R}$, acting on $\mathbb{R}$ by translation. The equation is solved by integrating: $y=\int a(t)dt + C$.
2. Linear homogeneous ODE $y' = a(t)y$. The group in this case is $\mathbb{R}^+$, acting on $\mathbb{R}$ by multiplication (notice that this group action is not transitive; the point $0 \in \mathbb{R}$ is preserved; this corresponds to the fact that $y = 0$ is always a solution of a homogeneous linear ODE). The Lie algebra is $\mathbb{R}$, and the ODE is “solved” by exponentiating the vector field, and integrating. Hence $y = C e^{\int a(t)dt}$ is the general solution. In fact, in the previous example, the Lie algebra of the group of translations is also identified with $\mathbb{R}$, and “exponentiating” is the identity map.
3. Linear inhomogeneous ODE $y' = a(t)y + b(t)$. The group in this case is the affine group $\mathbb{R}^+ \ltimes \mathbb{R}$ where the first factor acts by dilations and the second by translation. The affine group is not abelian, so one cannot “integrate” a vector field directly, but it is solvable: there is a short exact sequence $\mathbb{R} \to \mathbb{R}^+ \ltimes \mathbb{R} \to \mathbb{R}^+$. The image in the Lie algebra of the group of dilations is the term $a(t)y$, which can be integrated as before to give an integrating factor $e^{\int a(t)dt}$. Setting $z = ye^{-\int a(t)dt}$ gives $z' = y'e^{-\int a(t)dt} - a(t)ye^{-\int a(t)dt} = b(t)e^{-\int a(t)dt}$ which is an indefinite integral, and can be solved by a further integration. In other words, we do one integration to change the structure group from $\mathbb{R}^+ \ltimes \mathbb{R}$ to $\mathbb{R}$ (“integrating out” the group of dilations) and then what is left is an abelian structure group, in which we can do “ordinary” integration. This procedure works whenever the structure group is solvable; i.e. whenever there is a finite sequence $G=G_0,\cdots,G_n=0$ where each $G_i$ surjects onto an abelian group, with kernel $G_{i-1}$, and after finitely many steps, the last kernel is trivial.
4. Ricatti equation $y' = a(t)y^2 + b(t)y + c(t)$. In this case, it is well-known that the equation can blow up in finite time, and one does not obtain a group of transformations of $\mathbb{R}$, but rather a group of transformations of the projective line $\mathbb{RP}^1 = \mathbb{R} \cup \infty$; another point of view says that one obtains a pseudogroup of transformations of subsets of $\mathbb{R}$. The group in this case is the projective group $\text{PSL}(2,\mathbb{R})$, acting by projective linear transformations. Let $A(t)$ be a $1$-parameter family of matrices in $\text{PSL}(2,\mathbb{R})$, say $A(t)=\left( \begin{smallmatrix} u(t) & v(t) \\ w(t) & x(t) \end{smallmatrix} \right)$, with $A(0)=\text{id}$. Matrices act on $\mathbb{R}$ by fractional linear maps; that is, $Az = (uz + v)/(wz+x)$ for $z \in \mathbb{R}$. Differentiating $A(t)z$ at $t=0$ one obtains $(Az)'(0) = (u'z+v')-z(w'z+x') = w'z^2 + (u'-x')z + v'$ which is the general form of the Ricatti equation. Since the group $\text{PSL}(2,\mathbb{R})$ is not solvable, the Ricatti equation cannot be solved in terms of elementary functions and integrals. However, if one knows one solution $y=z(t)$, one can find all other solutions as follows. Do a change of co-ordinates, by sending the solution $z(t)$ “to infinity”; i.e. define $x = 1/(y-z)$. Then as a function of $x$, the Ricatti equation reduces to a linear inhomogeneous ODE. In other words, the structure group reduces to the subgroup of $\text{PSL}(2,\mathbb{R})$ fixing the point at infinity (i.e. the solution $z(t)$), which is the affine group $\mathbb{R}^+ \ltimes \mathbb{R}$. One can therefore solve for $x$, and by substituting back, for $y$.

The Ricatti equation is important for the solution of second order linear equations, since any second order linear equation $y'' = a(t)y' + b(t)y + c(t)$ can be transformed into a system of two first order linear equations in the variables $y$ and $y'$. A system of first order ODEs in $n$ variables can be described in terms of pseudogroups of transformations of (subsets of) $\mathbb{R}^n$. A system of linear equations corresponds to the structure group $\text{GL}(n,\mathbb{R})$, hence in the case of a $2\times 2$ system, to $\text{GL}(2,\mathbb{R})$. The determinant map is a homomorphism from $\text{GL}(2,\mathbb{R})$ to $\mathbb{R}^*$ with kernel $\text{SL}(2,\mathbb{R})$; hence, after  multiplication by a suitable integrating factor, one can reduce to a system which is (equivalent to) the Ricatti equation.

Having seen these examples, one naturally wonders whether there are any other interesting families of equations and corresponding Lie groups acting on $1$-manifolds. In fact, there are (essentially) no other examples: if one insists on (finite dimensional) simple Lie groups, then $\text{SL}(2,\mathbb{R})$ is more or less the only example. Perhaps this is one of the reasons why the theory of ODEs tends to appear to undergraduates (and others) as an unstructured collection of rules and tricks. Nevertheless, recasting the theory in terms of geometric structures has the effect of clearing the air to some extent.

Geometric structures on $1$-manifolds arise also in the theory of foliations, which may be seen as a geometric abstraction of certain kinds of PDE. Suppose $M$ is a manifold, and $\mathcal{F}$ is a codimension one foliation. The foliation determines local charts on the manifold in which the leaves of the foliation intersect the chart in the level sets of a co-ordinate function. In the overlap of two such local charts, the transitions between the local co-ordinate functions take values in some pseudogroup. For certain kinds of foliations, this pseudogroup might be analytically quite rigid. For example, if $\mathcal{F}$ is tangent to the kernel of a nonsingular $1$-form $\alpha$ on $M$, then integrating $\alpha$ determines a metric on the leaf space which is preserved by the co-ordinate transformations, and the pseudogroup is conjugate into the group of translations. There are also some interesting examples where the pseudogroup has no interesting local structure, but where structure emerges on a macroscopic scale, because of some special features of the topology of $M$ and $\mathcal{F}$. For example, suppose $M$ is a $3$-manifold, and $\mathcal{F}$ is a foliation in which every leaf is dense. One knows for topological reasons (i.e. theorems of Novikov and Palmeira) that the universal cover $\tilde{M}$ is homeomorphic to $\mathbb{R}^3$ in such a way that the pulled-back foliation $\tilde{\mathcal{F}}$ is topologically a foliation by planes. One important special case is when any two leaves of $\tilde{\mathcal{F}}$ are a finite Hausdorff distance apart in $\tilde{M}$. In this case, the foliation $\tilde{\mathcal{F}}$ is topologically conjugate to a product foliation, and $\pi_1(M)$ acts on the leaf space (which is $\mathbb{R}$) by a group of homeomorphisms. The condition that pairs of leaves are a finite Hausdorff distance away implies that there are intervals $I$ in the leaf space whose translates do not nest; i.e. with the property that there is no $g \in \pi_1(M)$ for which $g(I)$ is properly contained in $I$. Let $I^\pm$ denote the two endpoints of the interval $I$. One defines a function $Z:\mathbb{R} \to \mathbb{R}$ by defining $Z(p)$ to be the supremum of the set of values $g(I^+)$ over all $g \in \pi_1(M)$ for which $g(I^-) \le p$. The non-nesting property, and the fact that every leaf of $\mathcal{F}$ is dense, together imply that $Z$ is a strictly increasing (i.e. fixed-point free) homeomorphism of $\mathbb{R}$ which commutes with the action of $\pi_1(M)$. In particular, the action of $\pi_1(M)$ is conjugate into the subgroup $\text{Homeo}^+(\mathbb{R})^{\mathbb{Z}}$ of homeomorphisms that commute with integer translation. One says in this case that the manifold $M$ slithers over a circle; it is possible to deduce a lot about the geometry and topology of $M$ and $\mathcal{F}$ from this structure. See for example Thurston’s paper, or my book.

A third significant way in which geometric structures arise on circles is in the theory of conformal welding. Let $\gamma:S^1 \to \mathbb{CP}^1$ be a Jordan curve in the Riemann sphere. The image of the curve decomposes the sphere into two regions homeomorphic to disks. Each open disk region can be uniformized by a holomorphic map from the open unit disk, which extends continuously to the boundary circle. These uniformizing maps are well-defined up to composition with an element of the Möbius group $\text{PSL}(2,\mathbb{R})$, and their difference is therefore a coset in $\text{Homeo}^+(S^1)/\text{PSL}(2,\mathbb{R})$ called the welding homeomorphism. Conversely, given a homeomorphism of the circle, one can ask when it arises from a Jordan curve in the Riemann sphere as above, and if it does, whether the curve is unique (up to conformal self-maps of the Riemann sphere). Neither existence nor uniqueness hold in great generality. For example, if the image $\gamma(S^1)$ has positive (Hausdorff) measure, any quasiconformal deformation of the complex structure on the Riemann sphere supported on the image of the curve will deform the curve but not the welding homeomorphism. One significant special case in which existence and uniqueness is assured is the case that $\gamma(S^1)$ is a quasicircle. This means that there is a constant $K$ with the property that if two points $p,q$ are contained in the quasicircle, and the spherical distance between the two points is $d(p,q)$, then at least one arc of the quasicircle joining $p$ to $q$ has spherical diameter at most $Kd(p,q)$. In other words, there are no bottlenecks where two points on the quasicircle come very close in the sphere without being close in the curve. Welding maps corresponding to quasicircles are precisely the quasisymmetric homeomorphisms. A homeomorphism is quasisymmetric if for every sufficiently small interval in the circle, the image of the midpoint of the interval under the homeomorphism is not too far from being the midpoint of the image of the interval; i.e. it divides the image of the interval into two pieces whose lengths have a ratio which is bounded below and above by some fixed constant. Other classes of geometric structures can be detected by welding: smooth Jordan circles correspond to smooth welding maps, real analytic circles correspond to real analytic welding maps, round circles correspond to welding maps in $\text{PSL}(2,\mathbb{R})$, and so on. Recent work of  Eero Saksman and his collaborators has sought to find the correct idea of a “random” welding, which corresponds to the kinds of Jordan curves generated by stochastic processes such as SLE. In general, the precise correspondence between the analytic quality of $\gamma$ and of the welding map is given by the Hilbert transform.

This list of examples of geometric structures on $1$-manifolds is by no means exhaustive. There are many very special features of $1$-dimensional geometry: oriented $1$-manifolds have a natural causal structure, which may be seen as a special case of contact/symplectic geometry; (nonatomic) measures on $1$-manifolds can be integrated to metrics; connections on $1$-manifolds are automatically flat, and correspond to representations. It would be interesting to hear other examples, and how they arise in various mathematical fields.

I was in Stony Brook last week, visiting Moira Chas and Dennis Sullivan, and have been away from blogging for a while; this week I plan to write a few posts about some of the things I discussed with Moira and Dennis. This is an introductory post about the Goldman bracket, an extraordinary mathematical object made out of the combinatorics of immersed curves on surfaces. I don’t have anything original to say about this object, but for my own benefit I thought I would try to explain what it is, and why Goldman was interested in it.

In his study of symplectic structures on character varieties $\text{Hom}(\pi,G)/G$, where $\pi$ is the fundamental group of a closed oriented surface and $G$ is a Lie group satisfying certain (quite general) conditions, Bill Goldman discovered a remarkable Lie algebra structure on the free abelian group generated by conjugacy classes in $\pi$. Let $\hat{\pi}$ denote the set of homotopy classes of closed oriented curves on $S$, where $S$ is itself a compact oriented surface, and let $\mathbb{Z}\hat{\pi}$ denote the free abelian group with generating set $\hat{\pi}$. If $\alpha,\beta$ are immersed oriented closed curves which intersect transversely (i.e. in double points), define the formal sum

$[\alpha,\beta] = \sum_{p \in \alpha \cap \beta} \epsilon(p; \alpha,\beta) |\alpha_p\beta_p| \in \mathbb{Z}\hat{\pi}$

In this formula, $\alpha_p,\beta_p$ are $\alpha,\beta$ thought of as based loops at the point $p$, $\alpha_p\beta_p$ represents their product in $\pi_1(S,p)$, and $|\alpha_p\beta_p|$ represents the resulting conjugacy class in $\pi$. Moreover, $\epsilon(p;\alpha,\beta) = \pm 1$ is the oriented intersection number of $\alpha$ and $\beta$ at $p$.

This operation turns out to depend only on the free homotopy classes of $\alpha$ and $\beta$, and extends by linearity to a bilinear map $[\cdot,\cdot]:\mathbb{Z}\hat{\pi} \times \mathbb{Z}\hat{\pi} \to \mathbb{Z}\hat{\pi}$. Goldman shows that this bracket makes $\mathbb{Z}\hat{\pi}$ into a Lie algebra over $\mathbb{Z}$, and that there are natural Lie algebra homomorphisms from $\mathbb{Z}\hat{\pi}$ to the Lie algebra of functions on $\text{Hom}(\pi,G)/G$ with its Poisson bracket.

The connection with character varieties can be summarized as follows. Let $f:G \to \mathbb{R}$ be a (smooth) class function (i.e. a function which is constant on conjugacy classes) on a Lie group $G$. Define the variation function $F:G \to \mathfrak{g}$ by the formula

$\langle F(A),X\rangle = \frac {d}{dt}|_{t=0} f(A\text{exp}{tX})$

where $\langle \cdot,\cdot\rangle$ is some (fixed) $\text{Ad}$-invariant orthogonal structure on the Lie algebra $\mathfrak{g}$ (for example, if $G$ is reductive (eg if $G$ is semisimple), one can take $\langle X,Y\rangle = \text{tr}(XY)$). The tangent space to the character variety $\text{Hom}(\pi,G)/G$ at $\phi$ is the first cohomology group of $\pi$ with coefficients in $\mathfrak{g}$, thought of as a $G$ module with the $\text{Ad}$ action, and then as a $\pi$ module by the representation $\phi$. Cup product and the pairing $\langle\cdot,\cdot\rangle$ determine a pairing

$H^1(\pi,\mathfrak{g})\times H^1(\pi,\mathfrak{g}) \to H^2(\pi,\mathbb{R}) = \mathbb{R}$

where the last equality uses the fact that $\pi$ is a closed surface group; this pairing defines the symplectic structure on $\text{Hom}(\pi,G)/G$.

Every element $\alpha \in \pi$ determines a function $f_\alpha:\text{Hom}(\pi,G)/G \to \mathbb{R}$ by sending a (conjugacy class of) representation $[\phi]$ to $f(\phi(\alpha))$. Note that $f_\alpha$ only depends on the conjugacy class of $\alpha$ in $\pi$. It is natural to ask: what is the Hamiltonian flow on $\text{Hom}(\pi,G)/G$ generated by the function $f_\alpha$? It turns out that when $\alpha$ is a simple closed curve, it is very easy to describe this Hamiltonian flow. If $\alpha$ is nonseparating, then define a flow $\psi_t$ by $\psi_t\phi(\gamma)=\phi(\gamma)$ when $\gamma$ is represented by a curve disjoint from $\alpha$, and $\psi_t\phi(\gamma)= \text{exp} tF_\alpha(\phi)\phi(\gamma)$ if $\gamma$ intersects $\alpha$ exactly once with a positive orientation (there is a similar formula when $\alpha$ is separating). In other words, the representation is constant on the fundamental group of the surface “cut open” along the curve $\alpha$, and only deforms in the way the two conjugacy classes of $\alpha$ in the cut open surface are identified in $\pi$.

In the important motivating case that $G = \text{PSL}(2,\mathbb{R})$, so that one component of $\text{Hom}(\pi,G)/G$ is the Teichmüller space of hyperbolic structures on the surface $S$, one can take $f = 2\cosh^{-1}\text{tr/2}$, and then $f_\alpha$ is just the length of the geodesic in the free homotopy class of $\alpha$, in the hyperbolic structure on $S$ associated to a representation. In this case, the symplectic structure on the character variety restricts to the Weil-Petersson symplectic structure on Teichmüller space, and the Hamiltonian flow associated to the length function $f_\alpha$ is a family of Fenchel-Nielsen twists, i.e. the deformations of the hyperbolic structure obtained by cutting along the geodesic $\alpha$, rotating through some angle, and regluing. This latter observation recovers a famous theorem of Wolpert, connected in an obvious way to his formula for the symplectic form $\omega = \sum dl_\alpha \wedge d\theta_\alpha$ where $\theta$ is angle and $l$ is length, and the sum is taken over a maximal system of disjoint essential simple curves $\alpha$ for the surface $S$.

The combinatorial nature of the Goldman bracket suggests that it might have applications in combinatorial group theory. Turaev discovered a Lie cobracket on $\mathbb{Z}\hat{\pi}$, and showed that together with the Goldman bracket, one obtains a Lie bialgebra. Motivated by Stallings’ reformulation of the Poincaré conjecture in terms of group theory, Turaev asked whether a free homotopy class contains a power of a simple curve if and only if the cobracket of the class is zero. The answer to this question is negative, as shown by Chas; on the other hand, Chas and Krongold showed that a class $\alpha$ is simple if and only if $[\alpha,\alpha^3]$ is zero. Nevertheless, the full geometric meaning of the Goldman bracket remains mysterious, and a topic worthy of investigation.