On endomorphisms of the direct product of two free groups (Q6110519)

The paper under review describes the endomorphisms and automorphisms of the direct product \(F_n\times F_m\) of two free groups of finite ranks \(n,m\geq 2\). Moreover, it is proven that the Whitehead problems for endomorphisms, monomorphisms, and automorphisms of \(F_n\times F_m\) are all solvable. In other words, given two elements \(x,y\in F_n\times F_m\), one can decide whether there exists an endomorphism (resp. monomorphism, automorphism) \(\varphi\) of \(F_n\times F_m\) such that \(x\varphi = y\), and one can find such a \(\varphi\) by giving images of generators. Conditions are provided for an endomorphism \(\varphi\) of \(F_n\times F_m\) to have a finitely generated \textit{fixed subgroup} and \textit{periodic subgroup}, which are defined to be \[ \mathrm{Fix}(\varphi) = \{x\in F_n\times F_m \mid x\varphi = x\} \text{ and }\mathrm{Per}(\varphi) = \bigcup_{k\geq 1}\mathrm{Fix}(\varphi^k), \] respectively. The uniformly continuous endomorphisms are also described when \(F_n\times F_m\) is endowed with the product metric given by taking the prefix metric in each component.