Reidemeister classes in lamplighter type groups

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We prove that for any automorphism ϕ of the restricted wreath product Z₂ wr Zᵏ and Z₃ wr Z²ᵈ the Reidemeister number R(ϕ) is infinite, i.e. these groups have the property R∞. For Z₃ wr Z²ᵈ⁺¹ and Zₚ wr Zᵏ, where p>3 is prime, we give examples of automorphisms with finite Reidemeister numbers. So these groups do not have the property R∞. For these groups and Zₘ wr Z, where m is relatively prime to 6, we prove the twisted Burnside-Frobenius theorem (TBFTf): if R(ϕ)<∞, then it is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations fixed by the action [ρ] [ρ∘ϕ].