2022/10/02 by Evgenij Troitsky, Troitsky, Evgenij
Mathematics · Engineering · #Finite Group Theory Research #Advanced Algebra and Geometry #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2210.00591
Suppose, G is a residually finite group of finite upper rank admitting an automorphism φ with finite Reidemeister number R(φ) (the number of φ-twisted conjugacy classes). We prove that such G is soluble-by-finite (in other words, any residually finite group of finite upper rank, which is not soluble-by-finite, has the R_∞ property). This reduction is the first step in the proof of the second main theorem of the paper: suppose, G is a residually finite group of finite Prüfer rank and φ is its automorphism with R(φ)<∞; then R(φ) is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations of G, which are fixed points of the dual map \widehatφ:[ρ]↦ [ρ∘ φ] (i.e., we prove the TBFTf, the finite version of the conjecture about the twisted Burnside-Frobenius theorem, for such groups).