Generic countably infinite groups

TL;DR

This paper investigates the properties of a generic countably infinite group within a topological space, revealing that such groups are algebraically closed, simple, and possess specific structural features, with results applicable to broader model-theoretic contexts.

Contribution

It establishes that all group properties with the Baire property are either meager or comeager, and characterizes the typical structure of countably infinite groups, including the existence of a comeager elementary equivalence class.

Findings

The generic group is algebraically closed and simple.

The generic group is not finitely generated or locally finite.

In the Abelian case, the generic group is a divisible torsion group containing all finite Abelian groups.

Abstract

Countably infinite groups (with a fixed underlying set) constitute a Polish space $G$ with a suitable metric, hence the Baire category theorem holds in $G$. We study isomorphism invariant subsets of $G$, which we call group properties. We say that the generic countably infinite group is of property $P$ if $P$ is comeager in $G$. We prove that every group property with the Baire property is either meager or comeager. We show that there is a comeager elementary equivalence class in $G$ but every isomorphism class is meager. We prove that the generic group is algebraically closed, simple, not finitely generated and not locally finite. We show that in the subspace of Abelian groups the generic group is isomorphic to the unique countable, divisible torsion group that contains every finite Abelian group. We sketch the model-theoretic setting in which many of our results can be generalized. We briefly discuss a connection with infinite games.