A Lattice Point Problem on the Regular Tree

TL;DR

This paper investigates a lattice point counting problem on the regular tree by leveraging an analogy with hyperbolic geometry, providing asymptotic estimates for the number of automorphism images within expanding balls.

Contribution

It extends a classical hyperbolic lattice point problem to the setting of regular trees using geometric analogies, offering new asymptotic counting results.

Findings

Derived asymptotic formulas for lattice point counts on regular trees

Established a connection between hyperbolic plane problems and tree automorphisms

Provided a framework for counting automorphism images in expanding regions

Abstract

Heinz Huber (1956) considered the following problem on the the hyperbolic plane H. Consider a strictly hyperbolic subgroup of automorphisms on H with compact quotient, and choose a conjugacy class in this group. Count the number of vertices inside an increasing ball, which are images of a fixed point x in H under automorphisms in the chosen conjugacy class, and describe the asymptotic behaviour of this number as the size of the ball goes to infinity. We use a well-known analogy between the hyperbolic plane and the regular tree to solve this problem on the regular tree.