Limit groups as limits of free groups: compactifying the set of free groups

TL;DR

This paper introduces a topological framework for understanding limit groups as limits of free groups within a compact space of marked groups, providing new insights into their properties and relations.

Contribution

It establishes a topological approach to study limit groups, connecting them with ultraproducts and non-standard free groups, and offers new interpretations of existing results.

Findings

Limit groups are limits of free groups in a compact topology.

The topological perspective clarifies relations with ultraproducts and non-standard models.

New insights into the universal theory of free groups.

Abstract

We give a topological framework for the study of Sela's limit groups: limit groups are limits of free groups in a compact space of marked groups. Many results get a natural interpretation in this setting. The class of limit groups is known to coincide with the class of finitely generated fully residually free groups. The topological approach gives some new insight on the relation between fully residually free groups, the universal theory of free groups, ultraproducts and non-standard free groups.