Virtual Geometricity is Rare

TL;DR

This paper investigates the likelihood of multiwords in free groups being virtually geometric, showing through experiments and proof that this probability diminishes exponentially as multiword length increases.

Contribution

It provides the first empirical evidence and a rigorous proof that virtually geometric multiwords are exponentially rare in free groups.

Findings

Probability decays exponentially with multiword length

Empirical results support theoretical proof

Virtually geometric multiwords are extremely rare

Abstract

We present the results of computer experiments suggesting that the probability that a random multiword in a free group is virtually geometric decays to zero exponentially quickly in the length of the multiword. We then prove this fact.