We investigate fixed-point properties of automorphisms of groups similar to Richard Thompson’s group $$F$$ 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 $$R_\infty $$ R ∞ . Using the Bieri–Neumann–Strebel $$\varSigma $$ Σ -invariant and drawing from works of Gonçalves–Sankaran–Strebel and Zaremsky, we show that our tool applies to many $$F$$ F -like groups, including Stein’s group $$F_{2,3}$$ F 2 , 3 , cleary’s irrational-slope group $$F_\tau $$ F τ , the Lodha–Moore groups, and the braided version of $$F$$ F .

In 2013, Monod introduced a family of Thompson-like groups which provides natural counterexamples to the von Neumann-Day conjecture. We construct a characterization of conjugacy and an invariant and use them to compute centralizers in one group of this family.

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Monod introduced in [14] a family of Thompson-like groups which provides natural counterexamples to the von Neumann-Day conjecture. We construct a characterization of conjugacy and invariant and use them to compute centralizers in one group of this family.