On the asymptotics of visible elements and homogeneous equations in surface groups

TL;DR

This paper calculates the density of visible elements in surface groups and demonstrates that the probability of homogeneous equations having solutions approaches a value strictly between 0 and 1 as element lengths increase.

Contribution

It introduces methods to compute densities of visible elements in surface groups and analyzes the asymptotic probability of solutions to homogeneous equations.

Findings

Computed densities of visible elements in surface groups

Showed probability of solutions to homogeneous equations is neither 0 nor 1

Provided asymptotic analysis as element lengths grow

Abstract

Let $F$ be a group whose abelianization is $\Z^k$, $k\geq 2.$ An element of $F$ is called visible if its image in the abelianization is visible, that is, the greatest common divisor of its coordinates is 1. In this paper we compute three types of densities, annular, even and odd spherical, of visible elements in surface groups. We then use our results to show that the probability of a homogeneous equation in a surface group to have solutions is neither 0 nor 1, as the lengths of the right- and left-hand side of the equation go to infinity.