We give a definition of weakly sofic groups (w-sofic groups). Our definition is a rather natural extension of the definition of sofic groups where instead of the Hamming metric on symmetric groups we use general bi-invariant metrics on finite groups. The existence of non w-sofic groups is equivalent to a profinite topology property of products of conjugacy classes in free groups.

We study free profinite subgroups of free profinite semigroups of the same rank using, as main tools, iterated implicit operators, subword complexity and the associated entropy.

In this paper we identify a class of profinite groups (totally torsion free groups) that includes all separable Galois fields containing an algebraically closed subfield, and demonstrate it can be realized as inverse limit virtually finitely generated abelian (tfvfga) groups. We show by examples the condition is quite restrictive. particular, semidirect products are rarely totally free. The res...

Let R be a real closed field and L be a finite extension of R(t). We prove that Gal(L) ∼= Gal(R(t)) if L is formally real and Gal(L) is the free profinite group of rank card(R) if L is not formally real. MR Classification: 12E30 Directory: \Jarden\Diary\RealFree 3 April, 2007 * Research supported by the Minkowski Center for Geometry at Tel Aviv University, established by the Minerva Foundation....