Let $R$ be a compact boundary free riemann surface. Let $\Gamma$ be its fundamental group. Then $R\cong H/\Gamma$ where I abused notation for $\pi_1(R)\cong\Gamma$.
There is ambiguity coming from $\delta\in Aut(H),\Gamma'=\delta\Gamma\delta^{-1}$ with $H/\Gamma'$ defining same riemann surface. Let $A_i,B_i$ be generator of the fundamental group and I am going to abuse notation to denote $A_i,B_i$ the corresponding elements in $\Gamma$.
So we normalize the generator $A_i,B_i$ according the following. a) $B_i$ has repelling and attractive fixed points at $0,\infty$ respectively. b) $A_i$ has its attractive fixed point at $1$.
Q: How do I know there is always such a Fuchsian model with generator normalized according to $a),b)$ for each given genus $g>1$ riemann surface $R$?