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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