Lecture 11: Projective Structures on Surfaces

Let $\Sigma$ be a closed surface of genus at least 2, and $\Sigma_2$ a surface of genus $2$.

Let $\Gamma=\langle a,b,c,d\rangle$ be a discrete subgroup of $PSO(2,1)$. Then $\mathbb{H}^2/\Gamma\cong \Sigma_2$. The surface $\Sigma$ covers $\Sigma_2$ and if we let $\Gamma'\subset \Gamma$ be the subgroup corresponding to this cover then $\mathbb{H}^2/\Gamma'$ gives a convex projective structure on $\Sigma$

Let $\Omega\subseteq RP^2$ be an open properly convex set, let $\Gamma$ be a discrete subgroup of $PGL(\Omega)$, and $f\colon\Sigma\to \Omega/\Gamma$. Then the pair $(\Omega/\Gamma,f)$ is a marked convex projective structure on $\Sigma$. There is an equivalence relation on the collection of marked convex projective structures on $\Sigma$ by saying $(\Omega/\Gamma,f)\sim(\Omega'/\Gamma',f')$ if there is an element $g\in PGL_3(R)$ so that $gf$ is isotopic to $f'$

Teichmuller space $T(\Sigma)$ is contained in $\mathfrak{B}(\Sigma)$, the collection of equivalence classes of marked convex projective structures.

The character variety is $\chi(\Sigma)=Hom(\pi_1\Sigma,SL_3(R))/SL_3(R)$. There is a map $Hol\colon B(\Sigma)\to \chi(\Sigma)$, with $Hol(\Omega/\Gamma,f)=\left[f_*\colon \pi_1\Sigma\to\Gamma\subseteq SL_3(R)\right]$

Theorem:(Koszul, Goldman-Choi) Hol is a homeomorphism onto a connected component of $\chi(\Sigma)$.

Theorem:(Goldman) If $\Sigma$ has genus $g\geq 2$, then $\mathfrak{B}(\Sigma)$ is a cell of dimension $16g-16=-8\chi(\Sigma)$.

Outline:

1 Understand “$\mathfrak{B}(Pants)$”. It is an 8-dimensional cell.

2 If $\Sigma_1$ and $\Sigma_2$ are convex projective manifolds with “nice” boundary and their boundary geometry agrees, then they can be “convexly glued” in 2-dimensional ways.

Principal annuli

Let $D^+=\{\begin{pmatrix} e^{t_1} & & \\ & e^{t_2} & \\ & & e^{t_3}\end{pmatrix}\in SL_3(R)\mid t_3<t_2<t_1\text{ and } t_1+t_2+t_3=0\}$.

Let $\Delta=\{[e^{x_1},e^{x_2},e^{x_3}\mid x_1+x_2+x_3=0\}$. Let $1\not=\gamma=\begin{pmatrix} e^{t_1} & & \\ & e^{t_2} & \\ & & e^{t_3}\end{pmatrix}\in D^+$. Then choose $(a_1,a_2,a_3)$ such that $a_1t_1+a_2t_2+a_3t_3=0$. Then let $f_\gamma\colon\Delta\to R^+$ with $f_\gamma([e^{x_1},e^{x_2},e^{x_3}])=e^{a_1x_1+a_2x_2+a_3x_3}$. Then we have $f_\gamma(\gamma[e^{x_1},e^{x_2},e^{x_3})=f_\gamma([e^{x_1+t_1},e^{x_2+t_2},e^{x_3+t_3}])=f_\gamma([e^{x_1},e^{x_2},e^{x_3}])$.

$\Delta/\langle\gamma\rangle$ is foliated by the level sets of $f_\gamma$.

Let $A_d^\gamma=f^{-1}((d,\infty))/\langle\gamma\rangle$, which we call a principal annulus. Let $\Sigma=\Omega/\Gamma$ be properly convex with boundary. Let $\partial\subseteq \partial\Sigma$. We say that $\Omega/\Gamma$ has principal totally geodesic boundary if for each lift $\widetilde{\partial}$ of $\partial$ in $\widetilde{\Sigma}$, $D(\widetilde{\partial})$ is an embedding into an open segment in $RP^1\subseteq RP^2$ and $\rho(\pi_1\partial)$ conjugate into $D^+$.

The following picture illustrates the geometry of $\Omega$ when $\Omega/\Gamma$ has a principal boundary component and is useful to refer to when thinking about the following two lemmas.

Lemma: If $\Sigma\cong \Omega/\Gamma$ has principal totally geodesic boundary, then (after applying a projective transformation) $\Omega\subseteq\overline{\Delta}$ and $\Sigma$ contains an embedded principal annulus neighborhood of each principal boundary component.

Proof: We can always assume $\Omega\cap\overline{\Delta}\not=\varnothing$. By construction we know that $[e_1,e_3]\cap\overline{\Delta}\subseteq\partial\Omega$.

Let $\langle \gamma\rangle=\pi_1\partial$. For a generic $p\in \Omega^*$, we have $\gamma^np\to[e_3^*]\in\partial\Omega^*$ as $n\to\infty$ and $\gamma^{n}p\to[e_1^*]\in\partial\Omega^*$ as $n\to -\infty$. As a result we see that $[\ker e_1^\ast]$ and $[\ker e_3^\ast]$ are supporting hyperplanes. As a result we see that $\Omega$ is disjoint from these hyperplanes and thus $\Omega\subseteq \overline{\Delta}$.

In order to show that $\Sigma$ contains an embedded principal annulus we start by taking a regular neighborhood $N$ of $\partial$ in $\Sigma$ and lifting it to $\tilde N\subset \Omega$. Since $\partial$ is compact we can find a $\hat N\subset \tilde N$ which covers a principal annulus in $\Sigma$. Since $\hat N$ is contained in $\tilde N$ and $N$ is embedded we see that this principal annulus must be embedded.

Let $R$ be the reflection given in coordinates by $\begin{pmatrix}1 & & \\ & -1 & \\ & & 1\end{pmatrix}$

Lemma: If $g\in \Gamma\setminus\langle\gamma\rangle$, then $g R\Delta\subseteq \Delta\setminus \Omega$.

Proof: $\Omega\cap R\Delta=\varnothing$ so $\varnothing=g(\Omega\cap R\Delta)=\Omega\cap gR\Delta$. Given $g\in \Gamma\setminus\langle\gamma\rangle$ The set $gR\Delta$ is bounded by the three lines $g\tilde \partial$, $p_1:=g[\ker e_1^\ast]$ and $p_3=g[\ker e_3^\ast]$. The planes $p_1$ and $p_3$ are disjoint from $R\Delta$ and are thus correspond to points $p_1^\ast,p_3^\ast\in \partial \Omega^\ast\cap (R\Delta)^\ast$. The projective line $[p_1^\ast,p_3^\ast]$ is disjoint from $\Delta^\ast$. As a results we see that the pencil of hyperplanes containing $p_1$ and $p_3$ has its center in $\Delta$. Thus $gR\Delta\subset \Delta\setminus \Omega$.

Next time we will use this setup to show that properly convex surfaces with principal totally geodesic boundary components can be convexly glued together whenever their boundary geometry agrees.

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