Thompson's group $T$ is $\frac{3}{2}$-generated

TL;DR

This paper proves that Thompson's group T is -2-generated, meaning every nontrivial element can be paired with a conjugate of a specific element to generate the entire group, extending the -generation property to an infinite simple group.

Contribution

It establishes that Thompson's group T is -2-generated and identifies a specific element with a universal generating property for all nontrivial elements.

Findings

Thompson's group T is -2-generated.

Existence of a universal element ta in T.

Any nontrivial element of T can generate T with a conjugate of ta.

Abstract

Every finite simple group can be generated by two elements and, in fact, every nontrivial element is contained in a generating pair. Groups with this property are said to be $\frac{3}{2}$-generated, and the finite $\frac{3}{2}$-generated groups were recently classified. Turning to infinite groups, in this paper, we prove that the finitely presented simple group $T$ of Thompson is $\frac{3}{2}$-generated. Moreover, we exhibit an element $\zeta \in T$ such that for any nontrivial $\alpha \in T$, there exists $\gamma \in T$ such that $\langle \alpha, \zeta^\gamma \rangle = T$.