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

Abstract

We prove that N( f ) = |L( f )| for any continuous map f of a given infranilmanifold with Abelian holonomy group of odd order. This theorem is the analogue of a theorem of Anosov for continuous maps on nilmanifolds. We will also show that although their fundamental groups are solvable, the infranilmanifolds we consider are in general not solvmanifolds, and hence they cannot be treated using the techniques developed for solvmanifolds.

The Anosov theorem for infranilmanifolds with cyclic holonomy group

“…(b) In [Dekimpe et al 2006] we proved an analogue of Theorem 3.1: namely, that Anosov's theorem holds for infranilmanifolds with abelian holonomy group of odd order. In this case −1 is never an eigenvalue.…”

“…Based on Theorem 2.3 we described in [Dekimpe et al 2006] a class of maps on infranilmanifolds for which Anosov's theorem always holds. (We do not claim that such maps exist on all infranilmanifolds.)…”

“…Lemma 2.7 [Dekimpe et al 2006]. Let B, C ∈ M n ‫)ޒ(‬ be two real matrices such that BC = C B and B has only nonreal eigenvalues.…”

Nielsen zeta functions on infra-nilmanifolds up to dimension three

“…The following lemma will make our work a lot easier, since it provides us with a way to do all the computations in aff(H) by working on matrices. It is a generalized version of the representation found in [8].…”