Abstract: We construct braided versions sV_br 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.

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[16]. 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 F_∞. Another, more surprising, consequence is that (unlike F) the kernel of any map F_2,3→ ℤ is of type F_∞, 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.

Dissertation

Braided Brin-Thompson Groups (insert link here)