Abstract

This talk is Part II of an account of joint work with Meng-Che (Turbo) Ho and Julia Knight on the typicality of groups. In Part I, Julia Knight presented Gromov’s question about groups that inspired this work and its extension to algebraic varieties in general and gave some examples of behaviors that we encountered under various assumptions. In this talk, l discuss general conditions that imply some of the behaviors discussed in these examples. Our primary general result states that for a commutative generalized bijective variety and presentations with a single generator and single identity, the zero-one law holds and, furthermore, that the sentences with density 1 are those true in the free structure. The proof of this result requires a specialized version of Gaifman’s Locality Theorem that enables us to get a better bound on the complexity of the formulas of interest to us; I will sketch a proof of this version that uses saturation and then discuss the proof of our general result.