Combinatorial Growth in the Modular Group

TL;DR

This paper investigates the growth rates of reciprocal and closed geodesics on the modular orbifold, showing that as subsurfaces exhaust the orbifold, their geodesic growth rates converge to those of the entire orbifold.

Contribution

It establishes the convergence of growth rates for low lying reciprocal and closed geodesics to the full set on the modular orbifold.

Findings

Growth rate of low lying reciprocal geodesics converges to the full set

Growth rate of low lying closed geodesics converges to the full set

Results apply to an exhaustion of the modular orbifold by compact subsurfaces

Abstract

We consider an exhaustion of the modular orbifold by compact subsurfaces and show that the growth rate, in terms of word length, of the reciprocal geodesics on such subsurfaces (so named low lying reciprocal geodesics) converge to the growth rate of the full set of reciprocal geodesics on the modular orbifold. We derive a similar result for the low lying geodesics and their growth rate convergence to the growth rate of the full set of closed geodesics.