Generic algebraic properties in spaces of enumerated groups

TL;DR

This paper develops a topological framework using Polish topologies on spaces of countable enumerated groups, revealing that many group-theoretic phenomena are generic and connecting these to the word problem and model-theoretic forcing.

Contribution

It introduces a new topological approach to analyze generic properties in spaces of enumerated groups and links these to fundamental problems like the word problem and isomorphism classes.

Findings

Several group phenomena are shown to be generic in the introduced topologies.

A sufficient condition is provided for the non-existence of comeager isomorphism classes.

Connections are established between genericity, the word problem, and model-theoretic forcing.

Abstract

We introduce and study Polish topologies on various spaces of countable enumerated groups, where an enumerated group is simply a group whose underlying set is the set of natural numbers. Using elementary tools and well known examples from combinatorial group theory, combined with the Baire category theorem, we obtain a plethora of results demonstrating that several phenomena in group theory are generic. In effect, we provide a new topological framework for the analysis of various well known problems in group theory. We also provide a connection between genericity in these spaces, the word problem for finitely generated groups and model-theoretic forcing. Using these connections, we investigate the natural question: when does a certain space of enumerated groups contain a comeager isomorphism class? We obtain a sufficient condition that allows us to answer the question in the negative for the space of all enumerated groups and the space of left orderable enumerated groups. We document several open questions in connection with these considerations.