Braided Brin-Thompson Groups
View on arXiv. Submitted January 2021. (29 pages)
Abstract: We construct braided versions sVbr of the Brin-Thompson groups sV and prove that they are of type F∞. The proof involves showing that the matching complexes of colored arcs on surfaces are highly connected.
The BNSR-invariants of the Stein Group F2,3 (with Matthew Zaremsky)
View on arXiv. Submitted December 2020. (11 pages)
Abstract: The Stein group F2,3 is the group of orientation-preserving homeomorphisms of the unit interval with slopes of the form 2p3q (p,q∈ ℤ) and breakpoints in ℤ. This is a natural relative of Thompson's group F. In this paper we compute the Bieri-Neumann-Strebel-Renz (BNSR) invariants Σm(F2,3) of the Stein group for all m∈ ℕ. A consequence of our computation is that (as with F) every finitely presented normal subgroup of F2,3 is of type F∞. Another, more surprising, consequence is that (unlike F) the kernel of any map F2,3→ ℤ is of type F∞, even though there exist maps F2,3 → ℤ2 whose kernels are not even finitely generated. In terms of BNSR-invariants, this means that every discrete character lies in Σ(F2,3), but there exist (non-discrete) characters that do not even lie in Σ1(F2,3). To the best of our knowledge, F2,3 is the first group whose BNSR-invariants are known exhibiting these properties.