Thompson's group F is almost 3/2-generated

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