The BNSR-invariants of the Stein group 𝐹2,3

Abstract

Abstract The Stein group F 2 , 3 F_{2,3} is the group of orientation-preserving piecewise linear homeomorphisms of the unit interval with slopes of the form 2 p ⁢ 3 q 2^{p}3^{q} ( p , q ∈ Z p,q\in\mathbb{Z} ) and breakpoints in Z ⁢ [ 1 6 ] \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 ) \Sigma^{m}(F_{2,3}) of the Stein group for all m ∈ N m\in\mathbb{N} . A consequence of our computation is that (as with 𝐹) every finitely presented normal subgroup of F 2 , 3 F_{2,3} is of type F ∞ \operatorname{F}_{\infty} . Another, more surprising, consequence is that (unlike 𝐹) the kernel of any map F 2 , 3 → Z F_{2,3}\to\mathbb{Z} is of type F ∞ \operatorname{F}_{\infty} , even though there exist maps F 2 , 3 → Z 2 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 ∞ ⁢ ( F 2 , 3 ) \Sigma^{\infty}(F_{2,3}) , but there exist (non-discrete) characters that do not even lie in Σ 1 ⁢ ( F 2 , 3 ) \Sigma^{1}(F_{2,3}) . To the best of our knowledge, F 2 , 3 F_{2,3} is the first group whose BNSR-invariants are known exhibiting these properties.