The BNSR-invariants of the Stein group $F_{2,3}$

TL;DR

This paper computes the BNSR-invariants of the Stein group $F_{2,3}$, revealing novel properties about its subgroups and characters, and providing the first example of such invariants with these features.

Contribution

It provides the first complete computation of BNSR-invariants for $F_{2,3}$, uncovering unique subgroup and character properties not seen in similar groups.

Findings

Every finitely presented normal subgroup of $F_{2,3}$ is of type $ extrm{F}_ty$.

Kernel of any map $F_{2,3} obZ$ is of type $ extrm{F}_ty$.

Existence of non-discrete characters not in $ ext{Sigma}^1(F_{2,3})$.

Abstract

The Stein group $F_{2,3}$ is the group of orientation-preserving homeomorphisms of the unit interval with slopes of the form $2^p3^q$ ($p,q\in\mathbb{Z}$) and breakpoints in $\mathbb{Z}[\frac{1}{6}]$. This is a natural relative of Thompson's group $F$. In this paper we compute the Bieri-Neumann-Strebel-Renz (BNSR) invariants $\Sigma^m(F_{2,3})$ of the Stein group for all $m\in\mathbb{N}$. A consequence of our computation is that (as with $F$) every finitely presented normal subgroup of $F_{2,3}$ is of type $\textrm{F}_\infty$. Another, more surprising, consequence is that (unlike $F$) the kernel of any map $F_{2,3}\to\mathbb{Z}$ is of type $\textrm{F}_\infty$, even though there exist maps $F_{2,3}\to \mathbb{Z}^2$ whose kernels are not even finitely generated. In terms of BNSR-invariants, this means that every discrete character lies in $\Sigma^\infty(F_{2,3})$, but there exist (non-discrete) characters that do not even lie in $\Sigma^1(F_{2,3})$. To the best of our knowledge, $F_{2,3}$ is the first group whose BNSR-invariants are known exhibiting these properties.