Abstract:

The Tarski conjecture states that two f.g. free gps cannot be distinguished by the truthness of a given first order sentence. That is, a given sentence is true over any f.g. free gp if and only if it is true over the free gp on two generators. Given a sentence, we can naturally seek for f.g. gps that can be distinguished from free gps by the given sentence. An open question in this context states that given a sentence, the collection of f.g. gps that can be distinguished from free gps by the given sentence is negligible, that is, if the sentence is true over f.g. free gps, then it is true over almost all the f.g. gps. In our talk, we will present the formal ground of this question, and as much as time allows, we present some of the strategies that we plan to use in order to prove it.