The Structure of $\kappa$-Maximal Cofinitary Groups

TL;DR

This paper investigates the structure and properties of $oldsymbol{ ext{kappa}}$-maximal cofinitary groups for uncountable regular cardinals, revealing their orbit structure, realizability of partitions, and universality under certain set-theoretic assumptions.

Contribution

It establishes new structural results about $oldsymbol{ ext{kappa}}$-maximal cofinitary groups, including orbit bounds, realizability of partitions as orbits, and conditions for their universality.

Findings

Any $oldsymbol{ ext{kappa}}$-maximal cofinitary group has fewer than $oldsymbol{ ext{kappa}}$ orbits.

Partitions of $oldsymbol{ ext{kappa}}$ into fewer than $oldsymbol{ ext{kappa}}$ sets can be realized as orbits.

Existence of universal $oldsymbol{ ext{kappa}}$-maximal cofinitary groups under certain set-theoretic assumptions.

Abstract

We study $\kappa$-maximal cofinitary groups for $\kappa$ regular uncountable, $\kappa = \kappa^{<\kappa}$. Revisiting earlier work of Kastermans and building upon a recently obtained higher analogue of Bell's theorem, we show that: 1. Any $\kappa$-maximal cofinitary group has ${<}\kappa$ many orbits under the natural group action of $S(\kappa)$ on $\kappa$. 2. If $\mathfrak{p}(\kappa) = 2^\kappa$ then any partition of $\kappa$ into less than $\kappa$ many sets can be realized as the orbits of a $\kappa$-maximal cofinitary group. 3. For any regular $\lambda > \kappa$ it is consistent that there is a $\kappa$-maximal cofinitary group which is universal for groups of size ${<}2^\kappa = \lambda$. If we only require the group to be universal for groups of size $\kappa$ then this follows from $\mathfrak{p}(\kappa) = 2^\kappa$.