Virtually geometric words and Whitehead's algorithm

TL;DR

This paper investigates the concept of 'virtually geometric' words in group theory, demonstrating that certain words, such as 'bbaaccabc', do not possess this property, thus advancing understanding of their structural characteristics.

Contribution

The paper introduces the notion of virtually geometric words and proves that specific examples are not virtually geometric, providing new insights into their algebraic properties.

Findings

Certain words are proven not to be virtually geometric

The concept of virtually geometric words is formalized and explored

Results contribute to the understanding of Whitehead's algorithm and word properties

Abstract

Motivated by a question of Gordon and Wilton, we consider the question of which collections of words are "virtually geometric". In particular, we prove that some words (e.g. bbaaccabc) are not virtually geometric.