On fixed points and equalizers of injective endomorphisms of the free group at infinity

Abstract

We study equalizers and fixed points of monomorphisms of free groups at infinity. We show that the action of the equalizer of two monomorphisms on the regular points of the equalizer at infinity has finitely many orbits, showing that the equalizer at infinity is, in some sense, finitely generated and generalizing a previous result of Cooper about fixed points. We additionally show that it is decidable whether an automorphism of a free group has a nontrivial fixed point at infinity. The same result is shown for monomorphisms satisfying the condition of being almost length-increasing. We remark that almost length-increasing monomorphisms are generic among endomorphisms of a free group. We also prove that being almost length-increasing is a decidable condition. We end the paper with several open problems arising from this work.