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Twisted Burnside-Frobenius theory for discrete groups

Twisted Burnside-Frobenius theory for discrete groups,10.1515/CRELLE.2007.097,Journal Fur Die Reine Und Angewandte Mathematik,ALEXANDER FEL' SHTYN,Evg

Twisted Burnside-Frobenius theory for discrete groups   (Citations: 12)
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For a wide class of groups including polycyclic and finitely generated polynomial growth groups it is proved that the Reidemeister number of an automorphism f is equal to the number of finite-dimensional fixed points of the induced map f^ on the unitary dual, if one of these numbers is finite. This theorem is a natural generalization of the classical Burnside-Frobenius theorem to infinite groups. This theorem also has important consequences in topological dynamics and in some sense is a reply to a remark of J.-P. Serre. The main technical results proved in the paper yield a tool for a further progress.
Journal: Journal Fur Die Reine Und Angewandte Mathematik - J REINE ANGEW MATH , vol. 2007, no. 613, pp. 193-210, 2007
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    • ...Fel’shtyn and Troitsky [13] showed that this equality does hold for virtually polycyclic groups if one of the numbers in the equality is nite...

    Nic Kobanet al. A relationship between twisted conjugacy classes and the geometric inv...

    • ...of papers [25, 22, 32, 33, 95, 30, 28, 31, 29, 54] In the paper [35] we...
    • ...striction) (Definitions 2.3 and 2.5). This notion was used in [31] to...
    • ...finite groups are strongly twisted conjugacy separable groups [31, 35]...
    • ...gacy classes). Then G is a twisted conjugacy separable group [31, 35]...
    • ...(3) Examples of groups, which are not twisted conjugacy separable: HNN, Ivanov and Osin groups [31]...
    • ...is formed by the finite-dimensional representations [31, 35]...
    • ...Congruences for Reidemeister numbers[31]: Let φ : G → G be an...
    • ...[25, 32, 33, 95, 31, 29, 54] has yet to be verified...
    • ...This notion is closely related to the notion RP(φ) introduced in [31]...
    • ...Definition 2.4. [31] We say that a group G has the property RP if for...
    • ...Theorem 2.2. [31] Suppose, R(φ) < ∞. Then G is φ-conjugacy sep-...
    • ...Theorem 2.4. [31] Let F : → K be a morphism onto a finite group...
    • ...Theorem 2.6. [31] Any polycyclic-by-finite group is a strongly twisted...
    • ...Theorem 2.7. [31] Suppose, there exists a commutative diagram...
    • ...Theorem 2.8. [31] Let G be an φ-conjugacy separable group .Then...

    Alexander Fel'shtyn. New directions in Nielsen–Reidemeister theory

    • ...[12].) In particular, groups that are quasi-isometric to...

    DACIBERG GONCALVESet al. Twisted conjugacy classes in nilpotent groups

    • ...celebrated Burnside-Frobenius theorem [10, 12, 14, 11, 13]...

    Alexander Fel'shtynet al. Twisted conjugacy classes of automorphisms of Baumslag-Solitar groups

    • ...of the Burnside-Frobenius theorem [16, 12, 19, 20, 49, 18, 17], that is, to show...
    • ...conjecture [16, 19, 20, 49, 18, 17] has yet to be verified...

    ALEXANDER FEL' SHTYNet al. TWISTED CONJUGACY CLASSES IN SYMPLECTIC GROUPS, MAPPING CLASS GROUPS A...

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