Thompson-like groups, Reidemeister numbers, and fixed points

Abstract

We investigate fixed-point properties of automorphisms of groups similar to R. Thompson's group $F$. Revisiting work of Gonçalves-Kochloukova, we deduce a cohomological criterion to detect infinite fixed-point sets in the abelianization, implying the so-called property $R_{\infty }$. Using the BNS $\Sigma$-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 $F_{2,3}$, Cleary's $F_{\tau }$, the Lodha-Moore groups, and the braided version of $F$.

arXiv commentv3: 25 pages, 4 figures; Incorporated referees' suggestions, corrected Proposition 2.7, included new remarks. Final version, to appear in Geometriae Dedicata

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Cited by in corpus (1)

• Twisted conjugacy in soluble arithmetic groups