Thompson-like groups, Reidemeister numbers, and fixed points

Abstract:

We investigate fixed-point properties of automorphisms of groups similar to Richard Thompson’s group 𝐹. Revisiting work of Gonçalves and Kochloukova, we deduce a cohomological criterion to detect infinite fixed-point sets in the abelianization, implying the so-called property 𝑅∞. Using the Bieri–Neumann–Strebel Σ-invariant and drawing from works of Gonçalves–Sankaran–Strebel and Zaremsky, we show that our tool applies to many 𝐹-like groups, including Stein’s group 𝐹2,3, cleary’s irrational-slope group 𝐹𝜏, the Lodha–Moore groups, and the braided version of 𝐹.