TITLE: Gluing two hyperbolic surfaces along boundary components in terms of fuchsian representations.
QUESTION [2 upvotes]: Let $\Sigma:=\Sigma_g(\beta_1)$ and $\Sigma':=\Sigma_{g'}'(\beta_2)$ be two hyperbolic surfaces of genus $g$ and $g'$ with boundaries $\beta_1$ and $\beta_2$ of the same length.
Choose base points $p_1 \in \beta_1$ and $p_2 \in \beta_2$ and Fuchsian representation $\rho_1: \pi_1(\Sigma, p_1) \to \operatorname{PSL}(2, \mathbb{R})$ and $\rho': \pi_1(\Sigma', p_2) \to  \operatorname{PSL}(2, \mathbb{R})$.
Choose presentations
$$ \pi_1(\Sigma,p_1) = \langle  A_1, \dots A_g, B_1, \dots, B_g, \beta_1 \ | \  [A_1, B_1], \dots [A_g, B_g] \beta_1 =1 \rangle$$
$$ \pi_1(\Sigma',p_2) = \langle  A_1', \dots A_g', B_1', \dots, B_g', \beta_2 \ | \  [A_1', B_1'], \dots [A_g', B_g'] \beta_2 =1 \rangle$$
Since $\beta_1$ and $\beta_2$ are of the same lengths, there exists an isometry of $S^1$ sending $p_1 \mapsto p_2$. We use this isometry to glue $\Sigma_1$ to $\Sigma'$ along $\beta_1$ and $\beta_2$ and obtain new hyperbolic surface $\Sigma \cup_{\beta_1 } \Sigma'$.
Is there a natural way to describe the resulting Fuchsian representation $\rho'': \pi_1(\Sigma \cup_{\beta} \Sigma') \to \operatorname{PSL}(2, \mathbb{R})$?
I would appreciate a reference describing how gluing two hyperbolic surfaces translates to gluing their respective Fuchsian representations.

REPLY [2 votes]: Let me make some additional necessary assumptions for my answer. First, I'll assume that $\beta_1$ and $\beta_2$ are both geodesics. Second, I'll assume that they are oriented so that the gluing preserves orientation (which means one of them has the positive boundary orientation with respect to its surface, and the other has the negative boundary orientation).
Let's think of the oriented closed curve $\beta_i$ as representing an element of the fundamental group denoted $[\beta_i] \in \pi_1(\Sigma_i,p_i)$, i.e. the path homotopy class of $\beta_i$. Under the Fuchsian representation $\rho_i$, one obtains a hyperbolic element $\gamma_i = \rho_i[\beta_i] \in \text{PSL}(2,\mathbb R)$, whose axis is a hyperbolic geodesic $L_i$. Let's choose a base point $\tilde p_i \in L_i$ which is a lift of the base point $p$ (here I am using the developing map embedding of the universal covering space of $\Sigma_i$ into $\mathbb H^2$, with respect to which $\rho_i(\Sigma_i)$ acts as deck transformation group).
Since $\gamma_1$ and $\gamma_2$ are both orientation preserving hyperbolic isometries with the same translation length, they are conjugate elements of $\text{PSL}(2,\mathbb R)$. More specifically, there is a unique $\delta \in \text{PSL}(2,\mathbb R)$ such that $\delta(L_1)=L_2$ preserving orientation, and $\delta(\tilde p_1)=\tilde p_2$.
The Fuchsian representation $\rho''$ can now be described as follows. The restriction of $\rho''$ to $\pi_1(\Sigma_1,p_1)$ is equal to $\rho_1$. The restriction of $\rho''$ to $\pi_1(\Sigma_2,p_2)$ is equal to $\delta^{-1} \rho_2 \delta$. Both restrictions agree under the identification of $[\beta_1]=[\beta_2]$. Using the universal property for the amalgamated free product, it follows that $\rho''$ now extends to a representation $\pi_1(\Sigma_1 \cup_\beta \Sigma_2,p) \to \text{PSL}(2,\mathbb R)$.
There's still some work to do to prove that $\rho''$ is a representation that produces the glued up hyperbolic structure on the surface $\Sigma_1 \cup_\beta \Sigma_2$, but perhaps this is a good place to stop.
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