Free structures and limiting density

TL;DR

This paper investigates the properties of typical structures in algebraic varieties using limiting density, extending Gromov's question beyond groups to broader algebraic structures and identifying conditions for their elementary theories.

Contribution

It generalizes Gromov's question to arbitrary algebraic varieties and provides conditions under which free structures share the elementary theory of typical structures.

Findings

Different behaviors of limiting density are illustrated with examples.

Conditions are identified for the elementary theory of free structures to match that of typical structures.

The paper extends the concept of limiting density to a broad class of algebraic structures.

Abstract

Gromov asked what a typical (finitely presented) group looks like, and he suggested a way to make the question precise in terms of limiting density. The typical finitely generated group is known to share some important properties with the non-abelian free groups. We ask Gromov's question more generally, for structures in an arbitrary algebraic variety (in the sense of universal algebra), with presentations of a specific form. We focus on elementary properties. We give examples illustrating different behaviors of the limiting density. Based on the examples, we identify sufficient conditions for the elementary first-order theory of the free structure to match that of the typical structure; i.e., a sentence is true in the free structure iff it has limiting density 1.