Fixed points of endomorphisms of graph groups

It is shown, for a given graph group $G$, that the fixed point subgroup Fix$\,\varphi$ is finitely generated for every endomorphism $\varphi$ of $G$ if and only if $G$ is a free product of free abelian groups. The same conditions hold for the subgroup of periodic points. Similar results are obtained for automorphisms, if the dependence graph of $G$ is a transitive forest.Comment: 9 page

Twisted Burnside-Frobenius theory for endomorphisms of polycyclic groups

Let $R(\phi)$ be the number of $\phi$-conjugacy (or Reidemeister) classes of an endomorphism $\phi$ of a group $G$. We prove for several classes of groups (including polycyclic) that the number $R(\phi)$ is equal to the number of fixed points of the induced map of an appropriate subspace of the unitary dual space $\widehat G$, when $R(\phi)<\infty$. Applying the result to iterations of $\phi$ we obtain Gauss congruences for Reidemeister numbers. In contrast with the case of automorphisms, studied previously, we have a plenty of examples having the above finiteness condition, even among groups with $R_\infty$ property.Comment: 11 pages, v.2: small corrections, submitte

What an infra-nilmanifold endomorphism really should be

Infra-nilmanifold endomorphisms were introduced in the late sixties. They play a very crucial role in dynamics, especially when studying expanding maps and Anosov diffeomorphisms. However, in this note we will explain that the two main results in this area are based on a false result and that although we can repair one of these two theorems, there remains doubt on the correctness of the other one. Moreover, we will also show that the notion of an infra-nilmanifold endomorphism itself has not always been interpreted in the same way. Finally, we define a slightly more general concept of the notion of an infra-nilmanifold endomorphism and explain why this is really the right concept to work with

On periodic points of free inverse monoid endomorphisms

It is proved that the periodic point submonoid of a free inverse monoid endomorphism is always finitely generated. Using Chomsky's hierarchy of languages, we prove that the fixed point submonoid of an endomorphism of a free inverse monoid can be represented by a context-sensitive language but, in general, it cannot be represented by a context-free language.Comment: 18 page