A uniqueness theorem of reflectable deformations of a Fuchsian group

Abstract:

Let $\Gamma$ be a Fuchsian group of signature $(p,n,m;{\nu _1},{\nu _2}, \ldots ,{\nu _n})$; $2p - 2 + m + \sum \nolimits _{j = 1}^n {(1 - 1/{\nu _j}) > 0}$. Let ${I_1},{I_2}, \ldots ,{I_m}$ be a maximal set of inequivalent components of $\Omega \cap {\mathbf {\hat R}}$; $\Omega$ is the region of discontinuity and ${\mathbf {\hat R}}$ is the extended real line. Let $\phi$ be a quadratic differential for $\Gamma$. Let $f$ be a solution of the Schwarzian differential equation $Sf = \phi$. If $\phi$ is reflectable, $f$ maps each ${I_j}$ into a circle ${C_j}$. For each $\gamma \in \Gamma$ there is a Moebius transformation $\mathcal {X}(\gamma )$ such that $f \circ \gamma = \mathcal {X}(\gamma ) \circ f$. We prove that $\phi$ is determined by the homomorphism $\mathcal {X}$ and the circles ${C_1},{C_2}, \ldots ,{C_m}$.