Finite presentability of F(3/2)

Determine whether the Thompson-style group F(3/2) = G([0,1]; Z[1/6], <3/2>) is finitely presented; specifically, ascertain whether there exists a finite set of defining relations that, together with a finite generating set, presents this group.

Background

The group F(3/2) is a Bieri–Strebel type generalization of Thompson’s group, consisting of orientation-preserving piecewise-linear homeomorphisms of [0,1] with slopes restricted to powers of 3/2 and breakpoints in Z[1/6]. In this paper the authors prove that F(3/2) is finitely generated by two specific elements, addressing a longstanding question that arose since Stein’s paper of related groups.

Beyond finite generation, a central structural question is finite presentability, which requires a finite set of relations among generators. While the authors exhibit certain relations (see their formulas in the cited figure/section \Cref{xiv2}), they note that determining whether a finite set suffices remains unresolved and anticipate that any such relations could be long and complicated.