Rigid abelian groups and the probabilistic method

TL;DR

This paper introduces a probabilistic approach to constructing torsion-free abelian groups with specific endomorphism rings, simplifying previous methods by using random element selection to almost surely achieve the desired structure.

Contribution

It demonstrates that certain constructions of abelian groups can be simplified through randomization, ensuring the prescribed endomorphism ring with probability one.

Findings

Randomized construction simplifies traditional methods

Almost sure realization of prescribed endomorphism rings

Probabilistic approach reduces complexity of proofs

Abstract

The construction of torsion-free abelian groups with prescribed endomorphism rings starting with Corner's seminal work is a well-studied subject in the theory of abelian groups. Usually these construction work by adding elements from a (topological) completion in order to get rid of (kill) unwanted homomorphisms. The critical part is to actually prove that every unwanted homomorphism can be killed by adding a suitable element. We will demonstrate that some of those constructions can be significantly simplified by choosing the elements at random. As a result, the endomorphism ring will be almost surely prescribed, i.e., with probability one.