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The Stein group $F_{2,3}$ is the group of orientation-preservinghomeomorphisms 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 anatural relative of Thompson's group $F$. In this paper we compute theBieri-Neumann-Strebel-Renz (BNSR) invariants $\Sigma^m(F_{2,3})$ of the Steingroup for all $m\in\mathbb{N}$. A consequence of our computation is that (aswith $F$) 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 noteven finitely generated. In terms of BNSR-invariants, this means that everydiscrete 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 thebest of our knowledge, $F_{2,3}$ is the first group whose BNSR-invariants areknown exhibiting these properties.

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