Reidemeister classes, wreath products and solvability

2023/01/29 by Troitsky, Evgenij
#20C #20E45 #22D10 #37C25 #47H10 #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2301.12374

Abstract

Reidemeister (or twisted conjugacy) classes are considered in restricted wreath products of the form G\wr ℤ^k, where G is a finite group. For an automorphism φ of finite order (supposed to be the same for the torsion subgroup ⊕ G and the quotient ℤ^k) with finite number R(φ) of Reidemeister classes, this number is identified with the number of equivalence classes of finite-dimensional unitary irreducible representations of the product that are fixed by the dual homeomorphism \widehatφ (i.e. the so-called conjecture TBFT_f is proved in this case). For these groups and automorphisms, we prove the following conjecture: if a finitely generated residually finite group has an automorphism with R(φ)<∞ then it is solvable-by-finite (so-called conjecture R).

Related

• Twisted conjugacy in residually finite groups of finite Prüfer rank
• Reidemeister classes in lamplighter type groups
• Twisted conjugacy classes in residually finite groups
• Twisted Burnside theorem for type II₁ groups: an example
• Twisted conjugacy classes for polyfree groups