LAST TIME: Let $$Q$$ be the standard quadratic form on $$\mathbb R^{n + 1}$$ of signature $$(n, 1)$$, let $$\mathscr C$$ be the upper half of the cone $$Q^{-1}((-\infty, 0))$$, and let $$\mathcal H$$ be the upper sheet of the hyperboloid $$Q^{-1}(-1)$$. We have a map

\begin{align*} \Phi : \mathbb H^n &\longrightarrow{} (\mathbb H^n)^*, \\ x &\longmapsto{} [H_x], \end{align*}

where $$\mathbb H^n$$ is the projective model of hyperbolic space and

$H_x = x^{\perp_Q} = \{v \in \mathbb R^{n + 1} \mid \langle x, v \rangle_Q = 0\}.$

Recall that $$x$$ is the center of mass of $$\mathbb H^n$$ in the affine patch $$\mathbb R\mathrm P^n \setminus [H_x]$$.

TODAY, we’ll do a more general construction, starting with an arbirtrary fixed properly convex domain in projective space. The machinery we get out of it, along with John’s ellipsoid theorem (see below), will enable us to prove Benzecri’s compactness theorem.

Let $$V = \mathbb R^{n + 1}$$, fix a properly convex domain $$\Omega \subset \mathbb R\mathrm P^n$$, let $$\mathscr C$$ be the cone of $$\Omega$$, and let $$\mathscr C^*$$ be the dual cone. Define $$f : \mathscr C \to \mathbb R$$ by

$f(x) = \int_{\mathscr C^*} e^{-\varphi(x)} \, d\varphi,$

where $$d\varphi$$ is Lebesgue measure on $$V^*$$.

To compute $$f$$ explicity, we decompose $$\mathscr C^*$$ into slices as follows. For each $$x \in V$$ and each $$t > 0$$, let

$\mathscr C^*_x(t) = \{\varphi \in \mathscr C^* \mid \varphi(x) = t\}.$

Note that $$\mathscr C^*_x(t)$$ is the intersection of $$\mathscr C^*$$ with a hyperplane in $$V^*$$. We have

$\mathscr C^* = \bigcup_{t > 0} \mathscr C^*_x(t).$

Multiplication by $$t$$ gives a map $$\mathscr C^*_x(1) \to \mathscr C^*_x(t)$$. We have $$d\varphi = d\varphi_t \wedge dt$$ for some differential $$n$$-form $$d\varphi_t$$ on $$\mathscr C^*_x(t)$$. Then

$f(x) = \int^\infty_0 e^{-t} \int_{\mathscr C^*_x(t)} \, d\varphi_t \, dt = \int^\infty_0 e^{-t} t^n \int_{\mathscr C^*_x(1)} \, d\varphi_t \, dt = n! \, \mathrm{Vol}(\mathscr C^*_x(1)).$

It follows that

$f(\gamma x) = (\det \gamma)^{-1} f(x)$

for all $$\gamma \in \mathrm{GL}(\mathscr C)$$. The level sets of $$f$$ behave like $$\mathcal H$$ from last time.

The $$1$$-form $$\alpha = -d \log f = -df / f$$ maps $$\mathscr C$$ into $$\mathscr C^*$$. If $$v \in T_x\mathscr C \cong \mathbb R^{n + 1}$$, then

$df(x)(v) = -\int_{\mathscr C^*} \varphi(v) e^{-\varphi(x)} \, d\varphi,$

so that

$\alpha(x)(v) = \displaystyle\frac{\int_{\mathscr C^*} \varphi(v) e^{-\varphi(x)} \, d\varphi}{\int_{\mathscr C^*} e^{-\varphi(x)} \, d\varphi} > 0.$

The action of $$\mathrm{GL}(\mathscr C)$$ on $$\mathscr C$$ induces an action of $$\mathrm{GL}(\mathscr C)$$ on $$\mathscr C^*$$ given by $$\alpha(\gamma x) = \gamma \cdot \alpha(x)$$, where $$\gamma \cdot \alpha(x) \in V^*$$ is given by $$\gamma \cdot \alpha(x)(v) = \alpha(x)(\gamma^{-1}v)$$. It follows that $$\alpha$$ is $$\mathrm{GL}(\mathscr C)$$-equivariant. We calculate

$\alpha(x) = \displaystyle\frac{\int^\infty_0 e^{-t} t^{n + 1} \int_{\mathscr C^*_x(1)} \, d\varphi_t \, dt}{\int^\infty_0 e^{-t} t^n \int_{\mathscr C^*_x(1)} \, d\varphi_t \, dt} = (n + 1)\, \mathrm{Com}(\mathscr C^*_x(1)).$

There is a natural map $$\Omega \to \Omega^*$$ making the diagram

$\begin{matrix} \mathscr C & \smash{\stackrel{\alpha}{\longrightarrow}} & \mathscr C^* \\ \downarrow & & \downarrow \\ \Omega & \longrightarrow & \Omega^* \\ \end{matrix}$

commute. (Here the vertical maps are the quotient maps).

Remark. In Goldman’s notes, it is shown that $$f$$ has positive Hessian and $$\alpha$$ is a diffeomorphism.

Now we’ll define the duals of the maps $$f$$ and $$\alpha$$. As you’ll see, the definitions of these functions look similar to those of $$f$$ and $$\alpha$$.

Define $$f^* : \mathscr C^* \to \mathbb R$$ by

$f^*(\varphi) = \int_{\mathscr C} e^{\varphi(x)} \, dx = n! \, \mathrm{Vol}(\mathscr C_\varphi(1)),$

and define $$\alpha^* : \mathscr C^* \to \mathscr C$$ by

$\alpha^*(\varphi) = (n + 1) \, \mathrm{Com}(\mathscr C_\varphi(1)),$

where

$\mathscr C_\varphi(t) = \{x \in \mathscr C \mid \varphi(x) = t\}$

is the intersection of $$\mathscr C$$ with a hyperplane in $$V$$ for each $$\varphi \in V^*$$ and each $$t > 0$$. Now let

$\Phi = (\alpha^*)^{-1}.$

The map $$\Phi$$ sends each $$x \in \mathscr C$$ to the intersection $$H_x \cap \mathscr C^*$$, where $$H_x$$ is the unique hyperplane in $$V^*$$ such that $$H_x \cap \mathscr C^*$$ has center of mass at $$x$$.

Remark. In general, $$\alpha^* \circ \alpha$$ is not equal to the identity map, but it is if $$\Omega$$ is $$\mathbb H^n$$. Thus the preceeding construction is a generalization of the construction from last time.

Now we just need some more notation to state Benzécri’s compactness theorem.

Let $$\mathfrak C(\mathbb R\mathrm P^n)$$ denote the set of all properly convex domains in $$\mathbb R\mathrm P^n$$. Topologize $$\mathfrak C(\mathbb R\mathrm P^n)$$ by giving it the Hausdorff topology on closures. Let $$G = \mathrm{PGL}(n + 1)$$. The action of $$G$$ on $$\mathbb R\mathrm P^n$$ induces an action of $$G$$ on $$\mathfrak C(\mathbb R\mathrm P^n)$$. This action is not very nice in that the orbit space $$\mathfrak C(\mathbb R\mathrm P^n) / G$$ is not Hausdorff (remember the “tear drop in a triangle” example from class), but we can get a nicer action by considering pointed domains instead.

Let $$\mathfrak C_*(\mathbb R\mathrm P^n)$$ be the set of all ordered pairs $$(\Omega, p)$$, where $$\Omega \in \mathfrak C(\mathbb R\mathrm P^n)$$ and $$p \in \Omega$$. The topology on $$\mathfrak C(\mathbb R\mathrm P^n)$$ gives a topology on $$\mathfrak C_*(\mathbb R\mathrm P^n)$$, and the action of $$G$$ on $$\mathfrak C(\mathbb R\mathrm P^n)$$ gives an action of $$G$$ on $$\mathfrak C_*(\mathbb R\mathrm P^n)$$.

Theorem (Benzécri compactness). The action of $$G$$ on $$\mathfrak C_*(\mathbb R\mathrm P^n)$$ is proper and cocompact.

We’ll use the following two results to prove this theorem:

Fact. If $$G$$ acts properly on $$X$$ and $$K$$ is a compact subset of $$X$$, then

$G_K = \{g \in G \mid g(K) \cap K \neq \emptyset\}$

is a compact subset of $$G$$.

Theorem (John ellipsoid). Let $$C \subset \mathbb R^n$$ be a compact convex set with nonempty interior and center of mass at the origin. There exists a unique ellipsoid $$E$$ of maximal volume contained in $$C$$ and centered at the origin. Furthermore, $$E \subseteq C \subseteq n E$$.

By applying an affine map if necessary, we may assume that the John ellipsoid of $$C$$ is the unit ball $$B \subset \mathbb R^n$$ centered at the origin.

Let $$\mathfrak E_*(\mathbb R\mathrm P^n)$$ be the set of all ordered pairs $$(E, p)$$, where $$E \subset \mathbb R\mathrm P^n$$ is an ellipsoid and $$p \in E$$. Then $$G$$ acts transitively on pointed ellipsoids with point stabilizers isomorphic to $$\mathrm{PO}(n)$$, so $$G$$ acts properly and cocompactly on $$\mathfrak E_*(\mathbb R\mathrm P^n)$$.

Proof of Benzécri compactness. We define a map $$\varphi : \mathfrak C_*(\mathbb R\mathrm P^n) \to \mathfrak E_*(\mathbb R\mathrm P^n)$$ as follows. Given $$(\Omega, p)$$, let $$p^* = \Phi(p)$$. Then $$p$$ is the center of mass of $$\Omega$$, regarded as a subset of the affine patch $$\mathbb R\mathrm P^n \setminus [\ker p^*]$$. Now let $$E$$ be the John ellipsoid for $$\Omega$$ in these coordinates, and set $$\varphi(\Omega, \rho) = (E, \rho)$$.

The map $$\varphi$$ is continuous, well-defined, and $$G$$-invariant from what we proved earlier. The fibers of $$\varphi$$ are cocompact, since there are only compactly many domains $$\Omega$$ between a ball $$B$$ and $$nB$$. Therefore, the action of $$G$$ is proper and cocompact. $$\square$$

Corollary. If $$\Omega \subset \mathbb R\mathrm P^n$$ is a properly convex domain and $$\Gamma$$ is a discrete subgroup of $$\mathrm{PGL}(\Omega)$$, then the action of $$\Gamma$$ on $$\Omega$$ is properly discontinuous.

Corollary. If $$\Omega \subset \mathbb R\mathrm P^n$$ is a properly convex domain, $$G$$ is a subgroup of $$\mathrm{PGL}(\Omega)$$, and the action of $$G$$ on $$\Omega$$ has a fixed point, then $$G$$ is conjugate into $$\mathrm{PO}(n)$$.

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