New Zeta Functions of Reidemeister Type and the Twisted Burnside–Frobenius Theory

Abstract

We introduce new zeta functions related to an endomorphism ϕ of a discrete group Γ. They are of two types: counting numbers of fixed (ρ ~ ρ o ϕn) irreducible representations for iterations of ϕ from an appropriate dual space of Γ and counting Reidemeister numbers R(φn) of different compactifications. Many properties of these functions and their coefficients are obtained. In many cases, it is proved that these zeta functions coincide. The Gauss congruences for coefficients are proved. Useful asymptotic formulas for the zeta functions are found. Rationality is proved for some classes of groups, including those, which give also the first counterexamples simultaneously for TBFT (R(ϕ) = the number of fixed irreducible unitary representations) and TBFTf (R(ϕ) = the number of fixed irreducible unitary finite-dimensional representations) for an automorphism ϕ with R(ϕ) < 8.