Extending partial automorphisms and the profinite topology on free groups

Abstract

A class of structures C \mathcal {C} is said to have the extension property for partial automorphisms (EPPA) if, whenever C 1 C_1 and C 2 C_2 are structures in C \mathcal {C} , C 1 C_1 finite, C 1 ⊆ C 2 C_1\subseteq C_2 , and p 1 , p 2 , … , p n p_1,p_2,\dotsc ,p_n are partial automorphisms of C 1 C_1 extending to automorphisms of C 2 C_2 , then there exist a finite structure C 3 C_3 in C \mathcal {C} and automorphisms α 1 , α 2 , … , α n \alpha _1, \alpha _2,\dotsc ,\alpha _n of C 3 C_3 extending the p i 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 Zalesskiĭstating that a finite product of finitely generated subgroups is closed for this topology.