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

Abstract

The Stein group $F_{2,3}$ is the group of orientation-preserving homeomorphisms of the unit interval with slopes of the form $2^p{3}^q$ ($p,q\in \mathbb{Z}$) 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 ${\text{F}}_{\infty }$. Another, more surprising, consequence is that (unlike $F$) the kernel of any map $F_{2,3}\to \mathbb{Z}$ is of type ${\text{F}}_{\infty }$, 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.

11 pages

References in corpus (1)

• The BNSR-invariants of the Lodha-Moore groups, and an exotic simple group of type ${\text{F}}_{\infty }$