| Résume | We will consider metric reduced products of finite symmetric groups with respect to the normalized Hamming metric. Via an argument of Elek and Szabó, these groups are related to the study of soficity of groups, as initiated by Gromov and Weiss: a countable group is sofic precisely when it embeds into such a reduced product. We will discuss the possibilities of characterizing all existing isomorphisms between such universal sofic groups and of all automorphisms of such groups being inner. Here, in addition to the (asymptotic) behaviour of the groups Sym(n), the specific set theoretic assumptions in use become highly relevant. In particular, we will make use of a set theoretic lifting result for maps between metric reduced products that was shown in joint work with Alessandro Vignati to hold under the conjunction of Martin’s Axiom and the Open Coloring Axiom.
This is joint work with Andreas Thom. |
