On Extending Stallings 2-Cores of Diagram Groups to R. Thompson's Group T and Jones Subgroup of T

Abstract

We extend the list of known diagram groups which satisfy a conjecture of Guba and Sapir. Specifically, Guba and Sapir extended the notion of the Stallings foldings for free groups to diagram groups, and showed that given a subgroup H, the Stallings 2-core that it generates accepts a possibly larger subgroup of H. They conjectured that it was the smallest subgroup containing H and closed under taking components. Golan recently proved this conjecture for R. Thompson's group F, and we prove that it is true for the generalized Thompson groups Fn. We also extend the foldings to R. Thompson's group T, an annular diagram group, and prove a similar statement about the subgroup that a given 2-core in T accepts. Finally, in joint work with Yunxiang Ren, we analyze Jones' subgroup of T, which Jones showed encoded all oriented links. We provide a finite presentation as a group and a description as an annular diagram group, and show that the 2-core of this subgroup of T accepts exactly the same subgroup. We also show that Jones' subgroup of T coincides with its commensurator in T, and hence the corresponding unitary representation of T is irreducible.