# Infinite $\frac{3}{2}$-generated groups

## TL;DR

This paper proves that a broad class of infinite simple groups, including Thompson's groups and their generalizations, are $rac{3}{2}$-generated, meaning every nontrivial element is part of a generating pair.

## Contribution

The authors establish that all groups in the families $V_n$, $V_n'$, and $mV$ are $rac{3}{2}$-generated, expanding known examples of such groups beyond Tarski monsters.

## Key findings

- All $V_n$, $V_n'$, and $mV$ groups are $rac{3}{2}$-generated.
- Infinite noncyclic $rac{3}{2}$-generated groups are rare, with these being new examples.
- The paper raises open questions about the properties and classifications of $rac{3}{2}$-generated groups.

## Abstract

Every finite simple group can be generated by two elements, and Guralnick and Kantor proved that, moreover, every nontrivial element is contained in a generating pair. Groups with this property are said to be $\frac{3}{2}$-generated. Thompson's group $V$ was the first finitely presented infinite simple group to be discovered. The Higman--Thompson groups $V_n$ and the Brin--Thompson groups $mV$ are two families of finitely presented groups that generalise $V$. In this paper, we prove that all of the groups $V_n$, $V_n'$ and $mV$ are $\frac{3}{2}$-generated. As far as the authors are aware, the only previously known examples of infinite noncyclic $\frac{3}{2}$-generated groups are the pathological Tarski monsters. We conclude with several open questions motivated by our results.