Thompson-like groups, Reidemeister numbers, and fixed points

<p><strong>Abstract:</strong></p> <p>We investigate fixed-point properties of automorphisms of groups similar to Richard Thompson’s group ��. 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 ��∞. 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 ��-like groups, including Stein’s group ��2,3, cleary’s irrational-slope group ����, the Lodha–Moore groups, and the braided version of ��. </p>