On endomorphisms of the direct product of two free groups

TL;DR

This paper characterizes endomorphisms and automorphisms of the direct product of two free groups, providing conditions for fixed points, periodic points, and uniform continuity, and solving Whitehead problems for these groups.

Contribution

It offers a comprehensive description of endomorphisms and automorphisms of the direct product of free groups, including fixed point properties and Whitehead problem solutions.

Findings

Conditions for finitely generated fixed point subgroups

Solutions to Whitehead problems for these groups

Criteria for uniform continuity of endomorphisms

Abstract

We describe the endomorphisms of the direct product of two free groups of finite rank and obtain conditions for which the subgroup of fixed points is finitely generated and we do the same for periodic points. We also describe the automorphisms of such a group and solve the three Whitehead problems for automorphisms, monomorphisms and endomorphisms for groups in this class. We also obtain conditions for an endomorphism to be uniformly continuous when a suitable metric is considered and study infinite fixed points of the extension to the completion of the group.