Isomorphism types of definable (maximal) cofinitary groups

TL;DR

This paper demonstrates the existence of various definable maximal cofinitary groups with specific isomorphism types, including an arithmetic maximal cofinitary group of a particular form, answering questions about their structure.

Contribution

It provides a short proof that    always has an arithmetic cofinitary representation and constructs maximal cofinitary groups of new isomorphism types, including those not decomposing into free products.

Findings

   has an arithmetic cofinitary representation.

Constructed a maximal cofinitary group of isomorphism type   ( imes F) for any finite group F.

Answered an open question about the structure of definable maximal cofinitary groups.

Abstract

Kastermans proved that consistently $\bigoplus_{\aleph_1} \mathbb{Z}_2$ has a cofinitary representation. We present a short proof that $\bigoplus_{\mathfrak{c}} \mathbb{Z}_2$ always has an arithmetic cofinitary representation. Further, for every finite group $F$ we construct an arithmetic maximal cofinitary group of isomorphism type $(\ast_{\mathfrak{c}} \mathbb{Z}) \times F$. This answers an implicit question by Schrittesser and Mejak whether one may construct definable maximal cofinitary groups not decomposing into free products.