Reidemeister classes in lamplighter-type groups

Abstract

We prove that for any automorphism of the restricted wreath product and the Reidemeister number is infinite (the property ). For and , where p > 3 is prime, we give examples of automorphisms with finite Reidemeister numbers. So these groups do not have the property . For these groups and , where m is relatively prime to 6, we prove the twisted Burnside-Frobenius theorem (TBFTf): if , then it is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations fixed by the action .