The number of homomorphisms from the Hawaiian earring group

TL;DR

This paper establishes a dichotomy in the number of homomorphisms from the Hawaiian earring group to groups of size less than continuum, based on whether the target group is noncommutatively slender.

Contribution

It introduces a clear dichotomy for homomorphism counts from the Hawaiian earring group to small groups, linking it to the property of being noncommutatively slender.

Findings

Homomorphism count equals the size of G if G is noncommutatively slender.

Homomorphism count is 2^{2^{ ext{aleph}_0}} if G is not noncommutatively slender.

An example of a noncommutatively slender group with a nontrivial divisible element is provided.

Abstract

We show a dichotomy for groups of cardinality less than continuum. The number of homomorphisms from the Hawaiian earring group to such a group $G$ is either the cardinality of $G$ in case $G$ is noncommutatively slender, or the number is $2^{2^{\aleph_0}}$ in case $G$ is not noncommutatively slender. An example of a noncommutatively slender group with nontrivial divisible element is exhibited.