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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.

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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

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  • $\begingroup$ The isometry in the first paragraph is not quite unique. Within the class of orientation preserving isometries such an isometry is unique. In general, such an isometry is unique up to postcomposing by a reflection across the $\gamma_2$ line, or equivalently precomposing by a reflection across the $\gamma_1$ line. $\endgroup$ – Lee Mosher Jan 8 at 14:35

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