The BNSR-invariants of the Stein group F_2,3

2020/12/09 by Robert Spahn, Matthew C. B. Zaremsky, Spahn, Robert +1
Mathematics · #Advanced Combinatorial Mathematics #Advanced Operator Algebra Research #Geometric and Algebraic Topology #math.GR

paper · pdf · doi:10.48550/arxiv.2012.05000

11 pages

arxiv created 2020/12/09 · arxiv updated 2020/12/10

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∈ℤ) and breakpoints in ℤ[(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∈ℕ. A consequence of our computation is that (as with F) every finitely presented normal subgroup of F_2,3 is of type \textrmF_∞. Another, more surprising, consequence is that (unlike F) the kernel of any map F_2,3→ℤ is of type \textrmF_∞, even though there exist maps F_2,3→ ℤ² whose kernels are not even finitely generated. In terms of BNSR-invariants, this means that every discrete character lies in Σ^∞(F_2,3), but there exist (non-discrete) characters that do not even lie in Σ¹(F_2,3). To the best of our knowledge, F_2,3 is the first group whose BNSR-invariants are known exhibiting these properties.

Citations

• The BNSR-invariants of the Lodha-Moore groups, and an exotic simple group of type \textrmF_∞

Related

• The Σ-invariants of Thompson's group F via Morse theory
• The Sigma Invariants of Thompson's Group F
• Studying Stein's Groups as Topological Full Groups
• The BNSR-invariants of the Lodha-Moore groups, and an exotic simple group of type \textrmF_∞
• BNSR-Invariants of Surface Houghton Groups