Reidemeister and Nielsen zeta-functions

In this paper we formulate a rationality theorem for the Reidemeister and Nielsen zeta-functions modulo a normal subgroup of the fundamental group. We give conditions under which these zeta-functions coincide. We formulate a conjecture aboutentropy for the Reidemeister numbers. We show that the radius of convergence of the Nielsen zeta-function for an orientation-preserving homeomorphism f of a compact surface is an invariant of a three-dimensional manifold, the torus of the map f, and a special flow on it. In special cases we derive a functional equation for the Nielsen zeta-function. We give an example of a transcendental Nielsen zeta function.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 167, pp. 164–168, 1988.In conclusion the author expresses thanks to V. B. Piloginaya, V. G. Turaev, Boju Jiang, N. V. Ivanov for stimulating discussions and to D. Fried for sending preprints.