论文标题
Stein Group $ f_ {2,3} $的BNSR-Invariants
The BNSR-invariants of the Stein group $F_{2,3}$
论文作者
论文摘要
Stein Group $ f_ {2,3} $是单位间隔的一组同构的同构，其斜率为$ 2^p3^q $（$ p，q \ in \ mathbb {z} $），$ \ \ \ \ \ \ \ \ thebb {z} [z} [z} frac {\ frac {\ frac {1} $} $。这是汤普森集团$ f $的天然亲戚。在本文中，我们计算了bieri-neumann-strebel-renz（BNSR）不变性$σ^m（f_ {2,3}）$的所有$ m \ in \ mathbb {n} $。我们计算的结果是（与$ f $一样）$ f_ {2,3} $的每个有限呈现的正常子组为$ \ textrm {f} _ \ infty $。另一个更令人惊讶的结果是，（与$ f $不同）任何地图$ f_ {2,3} \ to \ mathbb {z} $的内核是类型的$ \ textrm {f} _ \ infty $，即使存在map $ f_ {2,3} \ to^toserers y k {2,3} \ to s z} $ n's Maps $ f_ {2,3} \ therers n's Kern;就bnsr-invariants而言，这意味着每个离散字符都在$σ^\ infty（f_ {2,3}）$中，但是存在（非散差）字符，甚至不在$σ^1（f_ {2,3}）中。据我们所知，$ f_ {2,3} $是第一个众所周知的BNSR-Invariants展示这些属性的组。
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 $Σ^m(F_{2,3})$ 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 $Σ^\infty(F_{2,3})$, but there exist (non-discrete) characters that do not even lie in $Σ^1(F_{2,3})$. To the best of our knowledge, $F_{2,3}$ is the first group whose BNSR-invariants are known exhibiting these properties.
