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Two examples related to the twisted Burnside-Frobenius theory for infinitely generated groups. (English. Russian original) Zbl 1479.20024

J. Math. Sci., New York 248, No. 5, 661-666 (2020); translation from Fundam. Prikl. Mat. 21, No. 5, 219-227 (2016).
Let \(\phi: G\to G\) be an automorphism of a (countable) discrete group. One has the \(\phi\)-twisted conjugation relation on \(G\) defined as \(x\sim y\) if there exists a \(g\in G\) such that \(y=gx\phi(g^{-1})\). The equivalence classes are called Reidemeister classes or \(\phi\)-twisted conjugacy classes. The cardinality of the set of all \(\phi\)-twisted conjugacy classes is the Reidemeister number \(R(\phi)\) of \(\phi\). One says that \(G\) has the property \(R_\infty\) if \(R(\phi)=\infty\) for every automorphism \(\phi\) of \(G\).
Let \(\hat G\) denote the unitary dual of \(G\) consisting of all equivalence classes of irreducible unitary representations of \(G\) and let \(\hat G_f\subset \hat G\) the subset consisting of classes of finite dimensional representations of \(G\). We have an action of \(\mathrm{Aut}(G)\) on \(\hat G,\) which stabilizes \(\hat G_f\), where \(\hat\phi: \hat G\to \hat G\) is defined as \([\rho]\mapsto [\rho\circ \phi]\) for \(\phi\in \mathrm{Aut}(G)\).
The classical Burnside-Frobenius theorem (TBFT) states that for a finite group \(G\), the set of isomorphism classes of finite dimensional irreducible representations are in bijection with the set of conjugacy classes of \(G\). It was observed by [A. Fel’shtyn and R. Hill, \(K\)-Theory 8, No. 4, 367–393 (1994; Zbl 0814.58033)] that if \(\phi:G\to G\) is an automorphism of a finite group \(G\), then \(\#\mathrm{Fix}(\hat\phi)=R(\phi)\). The conjecture \(\mathrm{TBFT}_f\) for an automorphism \(\phi\in \mathrm{Aut}(G)\) is that if \(R(\phi)<\infty\), then the number of fixed points of the \(\hat\phi:\hat G_f\to \hat G_f\) equals \(R(\phi)\).
The author considers two examples of countably generated groups \(G\) which are residually finite and so admit plenty of finite dimensional irreducible unitary representations. Certain automorphisms of these groups are used to test the validity of the conjecture \(\mathrm{TBFT}_f\). In the first example, the group is a free group \(F_\infty\) of countably infinite rank and in the second, a direct sum \(F^\infty:=\oplus_{n\in \mathbb Z}F_n\) where each \(F_n\) is a copy of a finite group \(F\).
In the case when \(G=F_\infty\), K. Dekimpe and D. Gonçalves [Bull. Lond. Math. Soc. 46, No. 4, 737–746 (2014; Zbl 1304.20051)] exhibited, for each \(n\ge 1\), plenty of automorphisms \(\phi_n:F_\infty\to F_\infty\) such that \(R(\phi_n)=n\). The author shows that \(\mathrm{TBFT}_f\) holds true for \(\phi_n\). He further shows that the fixed subgroups of these automorphisms have infinite index in \(F_\infty\).
In the case when \(G=F^\infty\), the author considers the right shift \(\phi ((x_j)_{j\in \mathbb Z}=(x_{j-1})_{j\in \mathbb Z}\). It is shown that \(R(\phi)\) equals \(\#F<\infty\). Taking \(F\) to be such that its centre is trivial, the author shows that \(\mathrm{TBFT}_f\) fails for \(\rho\).

MSC:

20E36 Automorphisms of infinite groups
20C15 Ordinary representations and characters
20E26 Residual properties and generalizations; residually finite groups
20E45 Conjugacy classes for groups
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References:

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