Inverse Limits of Finite Rank Free Groups

TL;DR

This paper characterizes the inverse limits of finite rank free groups, showing they are either finite rank free groups or a universal group, with connections to the Hawaiian earring's shape group.

Contribution

It establishes a classification of inverse limits of finite rank free groups and links them to the Hawaiian earring's shape group, introducing a new universal inverse limit.

Findings

Inverse limits are either finite rank free groups or a universal group.

The universal inverse limit is isomorphic to the Hawaiian earring's shape group.

An example of a homomorphic image that is neither free nor Hawaiian earring group.

Abstract

We will show that all inverse limits of finite rank free groups index by the natural numbers are isomorphic either to a finite rank free group or to a fixed universal group. In other words, any inverse system of finite rank free groups which is not equivalent to an eventually constant system has the universal group as its limit. This universal inverse limit is naturally isomorphic to the first shape group of the Hawaiian earring. We also give an example of a homomorphic image of a Hawaiian earring group which lies in the inverse limit of free groups but is neither a free group nor a Hawaiian earring group.