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Twisted Burnside theorem for type \(\text{II}_1\) groups: an example. (English) Zbl 1143.20026

Summary: The purpose of the present paper is to discuss the following conjecture of Fel’shtyn and Hill, which is a generalization of the classical Burnside theorem: Let \(G\) be a countable discrete group, \(\varphi\) its automorphism, \(R(\varphi)\) the number of \(\varphi\)-conjugacy classes (Reidemeister number), \(S(\varphi)=\#\text{Fix}(\widehat\varphi)\) the number of \(\varphi\)-invariant equivalence classes of irreducible unitary representations. If one of \(R(\varphi)\) and S\((\varphi)\) is finite, then it is equal to the other.
This conjecture plays a very important role in the theory of twisted conjugacy classes having a long history [see B. Jiang, Lectures on Nielsen fixed point theory. (1983; Zbl 0512.55003), A. Fel’shtyn, Dynamical zeta functions, Nielsen theory and Reidemeister torsion. (2000; Zbl 0963.55002)] and has very serious consequences in dynamics, while its proof needs rather fine results from functional and non-commutative harmonic analysis. It was proved for finitely generated groups of type I by the first two authors [Aspects of Mathematics E 37, 141-154 (2006; Zbl 1147.20036)].
In the present paper this conjecture is disproved for non-type I groups. More precisely, an example of a group and its automorphism is constructed such that the number of fixed irreducible representations is greater than the Reidemeister number. But the number of fixed finite-dimensional representations (i.e. the number of invariant finite-dimensional characters) in this example coincides with the Reidemeister number.
The directions for search of an appropriate formulation are indicated (another definition of the dual object).

MSC:

20F65 Geometric group theory
20E36 Automorphisms of infinite groups
20E45 Conjugacy classes for groups
55M20 Fixed points and coincidences in algebraic topology
57M05 Fundamental group, presentations, free differential calculus
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
20C15 Ordinary representations and characters
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