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A relationship between twisted conjugacy classes and the geometric invariants \(\Omega^n\). (English) Zbl 1223.20035
For a given finitely generated group \(G\), the invariants \(\Omega^n\), \(n\geq 1\), were defined by N. Koban [Topology Appl. 153, No. 12, 1975-1993 (2006; Zbl 1153.20041)] and are analogs of the Bieri-Neumann-Strebel-Renz invariants \(\Sigma^n\). In the paper under review the authors use succesfully these invariants to decide if certain groups have the \(R_\infty\) property, i.e. if for every automorphism \(\varphi\colon G\to G\) the number of \(\varphi\)-twisted conjugacy classes is infinite. They profit from the calculation of the invariants \(\Omega^n\) for certain groups.
The main result of the paper is Theorem 4.3: Let \(n\) be a positive integer, and let \(G\) be a group of type \(F_n\) with \(0<\#\Omega^n(G)<\infty\). Suppose that \(\Omega^n(G)\) contains only rational points.
(1) If \(\#\Omega^n(G)=1\), then \(G\) has property \(R_\infty\).
(2) If \(\#\Omega^n(G)=2\), then there exists a normal subgroup \(N\triangleleft\operatorname{Aut}(G)\) with \([\operatorname{Aut}(G):N]=2\) such that \(R(\varphi)=\infty\) for every \(\varphi\in N\).
As one application of the result above they prove Theorem 5.2: Let \(G=*_{i=1}^nA_i\) be a finite free product of \(n\) non-trivial freely indecomposable finitely generated groups, \(n\geq 2\). If one of the following conditions is satisfied then \(G\) has the property \(R_\infty\).
(1) Each \(A_i\) is finite.
(2) There exists \(j\), \(1\leq j\leq n\), such that \(A_j\in\mathcal O_1^m\) for \(i\neq j\), \(A_i\in\mathcal O_0^{k_i}\) with \(k_i\leq m\).
(3) The direct product \(\overline G=\prod A_i\) has property \(R_\infty\) and \(A_i\) is Abelian and non-isomorphic to \(\mathcal Z\) for some \(i\).
Here we denote by \(\mathcal O_i^k\) the family of finitely generated groups \(G\) such that \(\#\Omega^k(G)=i\).

MSC:
20F65 Geometric group theory
20E45 Conjugacy classes for groups
57M07 Topological methods in group theory
20E36 Automorphisms of infinite groups
55M20 Fixed points and coincidences in algebraic topology
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
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