Poincaré's theorem (geometry)

In mathematics , Poincaré's theorem gives a sufficient condition for a hyperbolic polygon to be the fundamental domain of a discrete group of isometries . It was proven by Henri Poincaré in 1882 and was fundamental to his work on the uniformization of Riemann surfaces .

Definitions

Let be a polygon in the hyperbolic plane . Let all edges of be provided with an orientation. Next, let us consider a group of isometrics of the hyperbolic plane. We say that two (oriented) edges and are paired with each other if there is one with . (The possibility is permitted.) An edge pairing of the polygon consists of a set of pairings in which each edge occurs exactly once as the starting edge and exactly once as the target edge. So for every pair of an edge pairing you have an isometric drawing . It is also required that the isometry assigned to the pair is the inverse of the isometry assigned to the pair , and that this applies to all edges . ${\displaystyle D}$ ${\displaystyle D}$ ${\displaystyle \Gamma }$ ${\displaystyle s}$ ${\displaystyle s^{\prime }}$ ${\displaystyle \gamma _{s}\in \Gamma }$ ${\displaystyle \gamma _{s}s=s^{\prime }}$ ${\displaystyle s=s^{\prime }}$ ${\displaystyle D}$ ${\displaystyle \gamma \in \Gamma }$ ${\displaystyle (s^{\prime },s)}$ ${\displaystyle \gamma _{s^{\prime }}}$ ${\displaystyle (s,s^{\prime })}$ ${\displaystyle \gamma _{s}\in \Gamma }$ ${\displaystyle \gamma _{s}(\mathrm {int} (D))\cap \mathrm {int} (D)=\emptyset }$ ${\displaystyle s}$

For a corner of there is a distinct oriented edge whose starting point is. Be . Let be the unique oriented edge with the starting point , and be . The iteration of this procedure must lead back to the starting corner after a finite number of steps . The cycle constructed in this way is called an elliptical cycle. ${\displaystyle v_{0}}$ ${\displaystyle D}$ ${\displaystyle s_{0}}$ ${\displaystyle v_{0}}$ ${\displaystyle v_{1}=\gamma _{s_{0}}v_{0}}$ ${\displaystyle s_{1}}$ ${\displaystyle v_{1}}$ ${\displaystyle v_{2}=\gamma _{s_{1}}v_{1}}$ ${\displaystyle v_{0}}$ ${\displaystyle v_{0}\to v_{1}\to v_{2}\to \ldots \to v_{0}}$

Poincaré's theorem

Let be a convex hyperbolic polygon with finitely many edges. One has an edge pairing in which no edge is paired with itself, and in which for each elliptical cycle the sum of the interior angles of the corners is of the form for a natural number . ${\displaystyle D}$ ${\displaystyle \textstyle {\frac {2\pi }{m_{j}}}}$ ${\displaystyle m_{j}}$

Then the edge pairings create a discrete group with fundamental domain . ${\displaystyle \Gamma }$ ${\displaystyle D}$

Generalizations

The three-dimensional version of Poincaré's theorem is called Poincaré's polyhedron substitution .

literature

• B. Maskit: On Poincaré's theorem for fundamental polygons , Adv. Math. 7, 219-230 (1971) online