Abstract

This talk is Part I of an account of joint work with Johanna Franklin and Meng-Che (Turbo) Ho.  Johanna Franklin’s talk is Part II.  Gromov asked, ``What is a typical group?’’  He was thinking of finitely presented groups, and he proposed an approach involving limiting density.  In 2013, I conjectured that for presentations with $n\geq 2$ generators and a single relator, the elementary first order sentences true in the typical group are those true in the free group.  The conjecture is still open, but there are partial positive results by Kharlampovich and Miasnikov, and by Ho and Logan.  In our joint work, Franklin, Ho, and I consider other algebraic varieties, in the sense of universal algebra, asking the analogue of Gromov’s question.  We have examples illustrating different possible behaviors.  For varieties with finitely many unary function symbols, we have a general result with conditions sufficient to guarantee that the analogue of the conjecture holds.  The proof uses a version of Gaifman’s Locality Theorem, plus ideas from random group theory.  Part I will describe Gromov’s original question and its extension to other algebraic varieties, with some examples.