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.
Braided Brin-Thompson Groups (insert link here)
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. In order to do so we develop the tools and definitions from algebraic topology and group theory, including results about some other Thompson-like groups. The main result, and the thesis as a whole, provides an infinite family of braided relatives of Thompson groups that are all of type F∞