I am quoting the following description from a paper,

"...every compact Riemann surface can be obtained as the quotient $\mathbb{C}/\Gamma$ where $\Gamma$ is a Schottky group. The Schottky group of a genus $g$ Riemann surface is a subgroup of $PSL(2,\mathbb{C})$ that is freely generated by $g$ loxodromic elements of $PSL(2,\mathbb{C})$. Let the $g$ generators of the Schottky group be $\{ L_i \}_{i=1}^g$. Then Mobius transformations map circles to circles and in particular for these loxodromic transformations, $2g$ disjoint circles $\{ C_i, C'_i\}_{i=1}^g$ can be chosen such that $L_i(C_i) = C_i'$. Under the quotient $\mathbb{C}/\Gamma$ these circles in $\mathbb{C}$ map to $g$ nontrivial elements of the fundamental group. The circles generate a maximal freely generated subgroup of the fundamental group. The remaining $g$ generators of the fundamental group are then obtained by paths that connect the pairs of circles.."

Can someone explain how are "loxodromic elements of $PSL(2,\mathbb{C})$ defined? How are they to be constructed given a Riemann surface?

How are the circles $C_{i=1, ..,g}$ to be chosen?

What is the action $L_j (C_i)$?

How to see the paths between the $C_i$s as becoming generators of teh fundamental group?

Is there a review paper available (best if online!) where this construction is explained? I haven't seen this in Riemann surface books I know of!

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