Estimating Nielsen numbers on infrasolvmanifolds (Q1204819)

The author's aim is the estimation of the Nielsen number \(N(f)\) for a selfmap \(f\) of an infrasolvmanifold from the Lefschetz number \(L(f)\) and other Lefschetz type numbers associated with \(f\). Here an infrasolvmanifold is an aspherical manifold whose fundamental group has a normal solvable subgroup of finite index. Hence such manifolds include as special cases nilmanifolds, infranilmanifolds and solvmanifolds, i.e. manifolds for which such estimations of \(N(f)\) exist. The main result states that if \(M\) is a compact infrasolvmanifold, then \(N(f)\geq| L(f)|\) for all maps \(f: M\to M\), and if, further, \(M\) is a compact infranilmanifold covered by a compact nilmanifold \(\widetilde M\), then \(N(f)\) can be computed from an expression which involves Lefschetz type numbers of lifts of \(f\) to \(\widetilde M\). Although this result concerns fixed points of \(f\), its statement and proof are obtained with the help of relations between Lefschetz and Nielsen type coincidence numbers, in particular those found recently by the author for coincidences of two maps between compact orientable solvmanifolds [\textit{C. McCord}, Topology Appl. 43, No. 3, 249-261 (1992; Zbl 0748.55001)]. The very clearly written paper contains welcome background material about the topology of infrasolvmanifolds and about existing results which relate \(N(f)\) and \(L(f)\) in other settings, as well as a study of Nielsen coincidence theory through lifts to finite regular covering spaces which is used in the proofs.