On endomorphisms of automatic groups (Q6630806)

Let \(F_{n}\) be a free group, \(\varphi \in \mathrm{Aut}(F_{n})\) and let \(\mathrm{Fix}(\varphi)=\{g \in F_{n} \mid g^{\varphi}=g \}\) be the subgroup of fixed points of \(\varphi\). \textit{S. M. Gersten} [Adv. Math. 64, 51--85 (1987; Zbl 0616.20014)] and \textit{D. Cooper} [J. Algebra 111, 453--456 (1987; Zbl 0628.20029)], using respectively graph-theoretic and topological approaches, proved that \(\mathrm{Fix}(\varphi)\) is always finitely generated. Subsequently \textit{M. Bestvina} and \textit{M. Handel} [Ann. Math. (2) 135, No. 1, 1--51 (1992; Zbl 0757.57004)], developed the theory of train tracks to prove that \(\mathrm{Fix}(\varphi)\) has rank at most \(n\). This line of research extended early to wider classes of groups (such as hyperbolic groups, Artin groups, lamplighter groups).\N\NIn the paper under review, the author deals with automatic groups, which constitute a broad class, which, in particular, contains direct products of hyperbolic groups. He extends the definition of the bounded reduction property to endomorphisms of automatic groups and finds conditions for it to hold. Furthermore, he studies endomorphisms with \(L\)-quasiconvex image and proves that those with finite kernel satisfy a synchronous version of the bounded reduction property. Finally, he uses the techniques he has developed to prove \(L\)-quasiconvexity of the equalizer of two endomorphisms under certain conditions.