Thompson's group F$F$ is almost 32$\frac{3}{2}$‐generated

Abstract

AbstractRecall that a group is said to be ‐generated if every nontrivial element of belongs to a generating pair of . Thompson's group was proved to be ‐generated by Donoven and Harper in 2019. It was the first example of an infinite finitely presented noncyclic ‐generated group. Recently, Bleak, Harper, and Skipper proved that Thompson's group is also ‐generated. In this paper, we prove that Thompson's group is “almost” ‐generated in the sense that every element of whose image in the abelianization forms part of a generating pair of is part of a generating pair of . We also prove that for every nontrivial element , there is an element such that the subgroup contains the derived subgroup of . Moreover, if does not belong to the derived subgroup of , then there is an element such that has finite index in .