The discrete Schwarz-Pick lemma for overlapping circles

Abstract:

Let P and $P’$ be circle packings in the hyperbolic plane such that they are combinatorically equivalent, neighboring circles in P overlap one another at some fixed angle between 0 and $\pi /2$ and the corresponding pairs of circles in $P’$ overlap at the same angle, and the radius for any boundary circle of P is less than or equal to that of the corresponding boundary circle of $P’$. In this paper we show that the radius of any interior circle of P is less than or equal to that of the corresponding circle in $P’$, and the hyperbolic distance between the centers of circles in P is less than or equal to the distance between the corresponding circles in $P’$. Furthermore, a single instance of finite equality in either of the above implies equality for all.