The "spread" of Thompson's group \(F\)

• Published in: arXiv.org 2024-02
• Main Author: Golan, Gili
• Format: Article
• Language: English
• Subjects: Integers; Recall
• ISSN: 2331-8422

Recall that a group \(G\) is said to be \(\frac{3}{2}\)-generated if every non-trivial element \(g\in G\) has a co-generator in \(G\) (i.e., an element which together with \(g\) generates \(G\)). Thompson's group \(V\) was proved to be \(\frac{3}{2}\)-generated by Donoven and Harper in 2019. It was the first example of an infinite finitely presented non-cyclic \(\frac{3}{2}\)-generated group. In 2022, Bleak, Harper and Skipper proved that Thompson's group \(T\) is also \(\frac{3}{2}\)-generated. Since the abelianization of Thompson's group \(F\) is \(\mathbb{Z}\), it cannot be \(\frac{3}{2}\)-generated. However, we recently proved that Thompson's group \(F\) is "almost" \(\frac{3}{2}\)-generated in the sense that every element of \(F\) whose image in the abelianization forms part of a generating pair of \(\mathbb{Z}^2\) is part of a generating pair of \(F\). A natural generalization of \(\frac{3}{2}\)-generation is the notion of spread. Recall that the spread of a group \(G\) is the supremum over all integers \(k\) such that every \(k\) non-trivial elements of \(G\) have a common co-generator in \(G\). The uniform spread of a group \(G\) is the supremum over all integers \(k\) for which there exists a conjugacy class \(C\subseteq G\) such that every \(k\) non-trivial elements of \(G\) have a common co-generator which belongs to \(C\). In this paper we study modified versions of these notions for Thompson's group \(F\).