Extending partial automorphisms and the profinite topology on free groups

Abstract

A class of structures $\mathcal {C}$ is said to have the extension property for partial automorphisms (EPPA) if, whenever $C_1$ and $C_2$ are structures in $\mathcal {C}$, $C_1$ finite, $C_1\subseteq C_2$, and $p_1,p_2,\dotsc ,p_n$ are partial automorphisms of $C_1$ extending to automorphisms of $C_2$, then there exist a finite structure $C_3$ in $\mathcal {C}$ and automorphisms $\alpha _1, \alpha _2,\dotsc ,\alpha _n$ of $C_3$ extending the $p_i$. We will prove that some classes of structures have the EPPA and show the equivalence of these kinds of results with problems related with the profinite topology on free groups. In particular, we will give a generalisation of the theorem, due to Ribes and ZalesskiÄ­stating that a finite product of finitely generated subgroups is closed for this topology.