The Nielsen numbers of virtually unipotent maps on infra-nilmanifolds

Abstract

An a½ne di¨eomorphism of an infra-nilmanifold M (with universal cover L) is virtually unipotent if and only if it lifts to an a½ne automorphism of L whose linear part has only eigenvalues of absolute value one. A continuous self-map of M which is homotopic to a virtually unipotent a½ne di¨eomorphism of M is referred to as a virtually unipotent map of M. In particular, we note that homotopically periodic self-maps of M are virtually unipotent.The main result of this paper is that, for each virtually unipotent map f of an infranilmanifold M, the Lefschetz number L f and the Nielsen number N f coincide. This nicely generalizes the analogous result of S. Kwasik and K. B. Lee, as presented in [11], for homotopically periodic maps of infra-nilmanifolds.

The Nielsen and Reidemeister numbers of maps on infra-solvmanifolds of type (R)

“…Therefore, for those classes of maps on infra-nilmanifolds for which Anosov relation N (f ) = |L(f )| holds [40,46,8] and for those classes of infranilmanifolds for which Anosov relation N (f ) = |L(f )| holds for ALL maps [1,7,8,9], the Nielsen zeta functions and the Reidemeister zeta functions are rational functions.…”

“…Remark 10.2. Recall [46,Lemma 3.6], which states that if an affine diffeomorphism f on an infra-nilmanifold M is homotopic to a virtually unipotent affine diffeomorphism on M , then f is virtually unipotent. However, the above example shows that this statement is not true in general for infrasolvmanifolds of type (R).…”

Bieberbach groups with multiplicity-free rational holonomy representation

“…Let us now recall the notion of a virtually unipotent map as introduced in [9] and discussed in more detail in [8]. …”

The anosov theorem for infranilmanifolds with an odd-order abelian holonomy group

“…Firstly, one can search classes of maps for which the relation holds for a specific type of manifold. For instance, Kwasik and Lee proved in [10] that the Anosov theorem holds for homotopic periodic maps of infranilmanifolds and in [14] Malfait did the same for virtually unipotent maps of infranilmanifolds. Secondly, one can look for classes of manifolds, other than nilmanifolds, for which the relation holds for all continuous maps of the given manifold, as was established by Keppelmann and McCord for exponential solvmanifolds (see [8]).…”