The $R_{\infty}$ property for infra-nilmanifolds

Karel Dekimpe, Bram de Rock, Pieter Penninckx

DOI: http://dx.doi.org/10.12775/TMNA.2009.047

Abstract


In this paper, we investigate the finiteness of the Reidemeister number $R(f)$
of a selfmap $f\colon M\to M$ on an infra-nilmanifold $M$.
We show that the Reidemeister number of an Anosov diffeomorphism on
an infra-nilmanifold is always finite.
A manifold $M$ is said to have the $R_\infty$ property if $R(f)=\infty$
for every homeomorphism $f\colon M\to M$.
We show that every non-orientable generalised Hantzsche-Wendt manifold
has the $R_\infty$ property.
For an orientable Hantzsche-Wendt manifold $M$, we formulate a criterion,
in terms of an associated graph, for $M$ to have the $R_\infty$ property.

Keywords


Reidemeister number; infra-nilmanifolds; Hantzsche-Wendt manifolds

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