Fixed points of endomorphisms of graph groups. (Q2869865)

For a group \(G\) let \(\Aut(G)\) denote its automorphism group and let \(\mathrm{End}(G)\) denote its endomorphism monoid and, given \(\varphi\in\mathrm{End}(G)\) let \(\mathrm{Fix}(\varphi)\) denote the set of fixed points of \(\varphi\) (i.e., those \(g\in G\) such that \(g\varphi=g\)) and let \(\mathrm{Per}(g)\) denote the set of all periodic points of \(\varphi\) (i.e., those \(g\in G\) such that \(g\varphi^n=g\) for some \(n\geq 1\)). \textit{S. M. Gersten} [Adv. Math. 64, 51-85 (1987; Zbl 0616.20014)] proved that if \(G\) is the automorphism group of a free group of finite rank then \(\mathrm{Fix}(G)\) is finitely generated. That result has been generalized in several directions and this article continues that theme of research.NEWLINENEWLINE Let \(\Gamma(A,I)\) be a graph with vertex set \(A\) and edge set \(I\). The graph group (or right angled Artin group) \(G(A,\Gamma)\) is the quotient of the free group \(F_A\) by the normal closure of \(\{[a,b]\mid (a,b)\in I\}\) in \(F_A\). This article considers which groups \(G(A,I)\) admit finitely generated fixed point and finitely generated periodic point subgroups for every \(\varphi\in\Aut(G)\) and for every \(\varphi\in\mathrm{End}(G)\). A complete answer is obtained for \(\mathrm{End}(G,A(I))\), namely that each of the following are equivalent: (i) \(\mathrm{Fix}(\varphi)\) is finitely generated for each \(\varphi\in\mathrm{End}(G(A,I))\), (ii) \(\mathrm{Per}(\varphi)\) is finitely generated for each \(\varphi\in\mathrm{End}(G(A,I))\), (iii) \(I\cup\{(a,a)\mid a\in A\}\) is transitive, (iv) \(\Gamma(A,I)\) is a disjoint union of complete graphs, (v) \(G(A,I)\) is a free product of finitely many free Abelian groups of finite rank. For \(\phi\in\mathrm{End}(\varphi)\) a partial answer is obtained, namely that in the case that \(\Gamma(A,I)\) is a transitive forest (that is, if it has no induced subgraph that is a 4-cycle or a path of length 3) then in the first result \(\mathrm{End}(G(A,I))\) may be replaced by \(\Aut(G(A,I))\). Examples are given that show that if \(\Gamma(A,I)\) is not a transitive forest then it may be that \(\mathrm{Fix}(\varphi)\), \(\mathrm{Per}(\varphi)\) are both finitely generated and it may be that neither is.