Effective equidistribution of circles in the limit sets of Kleinian groups
Wenyu Pan - Mathematics Department, Yale University, New Haven, CT 06520, United States (email) Abstract: Consider a general circle packing $\mathfrak{P}$ in the complex plane $\mathbb{C}$ invariant under a Kleinian group $\Gamma$. When $\Gamma$ is convex cocompact or its critical exponent is greater than 1, we obtain an effective equidistribution for small circles in $\mathfrak{P}$ intersecting any bounded connected regular set in $\mathbb{C}$; this provides an effective version of an earlier work of Oh-Shah [12]. In view of the recent result of McMullen-Mohammadi-Oh [6], our effective circle counting theorem applies to the circles contained in the limit set of a convex cocompact but non-cocompact Kleinian group whose limit set contains at least one circle. Moreover, consider the circle packing $\mathfrak{P}(\mathfrak{T})$ of the ideal triangle attained by filling in largest inner circles. We give an effective estimate to the number of disks whose hyperbolic areas are greater than $t$, as $t\to0$, effectivizing the work of Oh [10].
Keywords: Circle packing, Kleinian group.
Received: November 2015; Revised: October 2016; Available Online: February 2017. |