I have been working with Ricci flow in the euclidean and hyperbolic space but have been having considerable trouble determining how to generate a universal covering space for complex hyperbolic meshes.
Given a fundamental domain which is embedded into the Poincaré disk; I am unsure how to go about determining multiple levels of the UCS.
The primary issue that I am having is the inability to understand how to programmically go about using complex Möbius generating functions.
Any help understanding the following would be great help:
Let $\{ρ(a_k),ρ(a_k^{−1})\} ⊂ ∂ D$ be two boundary curve segments. We want to find a Möbius transformation $β_k$, such that $β_k(ρ(a^{−1}_k)) = ρ(a_k)$. Let their starting and ending vertices be $∂ρ(a^{−1}_k) = \{q_0, p_0\}$ and $∂ρ(a_k) = \{p_1, q_1\}$, then the Möbius transformation $β_k$ maps $(p_0, q_0)$ to $(p_1, q_1)$. $β_k$ is the Fuchsian generator corresponding to $b_k$. Similarly, we can compute $α_k$ which maps $ρ(b_k)$ to $ρ(b^{−1}_k)$. Therefore, we can compute a set of canonical Fuchsian group generators $\{α_1,β_1,α_2,β_2,\ldots,α_g,β_g\}$.