Orbital counting of curves on algebraic surfaces and sphere packings

TL;DR

This paper connects Apollonian circle packings with algebraic surface automorphisms, analyzing the growth of degrees of curves under group actions using orbit counting techniques.

Contribution

It realizes the Apollonian group as automorphisms of algebraic surfaces and applies orbit counting to study degree growth asymptotics.

Findings

Realized Apollonian group as surface automorphisms

Analyzed degree growth of curves under automorphism groups

Applied orbit counting to algebraic surface automorphisms

Abstract

We realize the Apollonian group associated to an integral Apollonian circle packings, and some of its generalizations, as a group of automorphisms of an algebraic surface. Borrowing some results in the theory of orbit counting, we study the asymptotic of the growth of degrees of elements in the orbit of a curve on an algebraic surface with respect to a geometrically finite group of its automorphisms.