Atlas for hyperbolic pair of pants I was reading about Teichmüller spaces and how you give a hyperbolic structure to a pair of pants via glueing two right angled hexagons, but I wanted to know if there was an explicit way to describe neighborhoods where is glued or a formula for translation maps between the "faces".
P.S. I´ve been looking for books about hyperbolic geometry, but haven´t find any one that I really like. Would appreciate some suggestions.
 A: The basic fact is, in hyperbolic plane, if $\gamma_1, \gamma_2$ are geodesics, $p_1, q_1, \in\gamma_1$  and $p_2,q_2\in\gamma_2$ and $d(p_1, q_1)=d(p_2, q_2)$, then there is a unique isometry that moves $\gamma_1$ to $\gamma_2$, $p_1$ to $p_2$, $q_1$ to $q_2$. Therefore a domain with a geodesic boundary piece can be glued to 
another domain with a geodesic boundary piece of the same length, the new combined surface is smooth, with constant curvature $-1$. 
Now take $2$ identical regular hexagons so that all angles are right angle, all sides are equal to $L$. Label the 6 sides of the first hexagon by $S_1, S_2, ..., S_6$, going counterclockwise; Label the 6 sides of the second hexagon by $T_1, T_2, ..., T_6$, going clockwise. Glue $S_1$ to $T_1$, $S_3$ to $T_3$, $S_5$ to $T_5$, you get a surface that is a sphere with three holes, the first hole has boundary $S_2\cup T_2$, the second hole has boundary $S_4\cup T_4$, the third hole has boundary $S_6\cup T_6$. Since the hexagon angles are all right angle, these three boundary pieces are all smooth circles. 
For further readings,, see
https://www.amazon.com/Lectures-Hyperbolic-Geometry-Universitext-Benedetti/dp/354055534X/ref=sr_1_16?keywords=hyperbolic+manifolds&qid=1578428703&sr=8-16
