Uniformity of the Meager Ideal and Maximal Cofinitary Groups

Abstract

We prove that every maximal cofinitary group has size at least the cardinality of the smallest non-meager set of reals. We also provide a consistency result saying that the spectrum of possible cardinalities of maximal cofinitary groups may be quite arbitrary.

Definable sets of generators in maximal cofinitary groups

“…A systematic study of mad permutation families was suggested by S. Thomas (see [14]) and many results have been obtained (see, e.g. [3,7,14,17], etc.). As to the relationship between maximal cofinitary permutation groups and mad permutation families in Sym(N), some preliminary study on set theoretic aspects of the two has been done in [15,19].…”

“…A group G Sym(N) is cofinitary or sharp if g is cofinitary for all g ∈ G \ {id}. For a discussion of different aspects of cofinitary groups, our reader can consult the well-written survey paper [4] by P. Cameron. Various research has been done concerning the structure of maximal cofinitary groups (see, e.g., [1][2][3]6,12,13,16,17], etc.). Most of the research has been focused on combinatorial properties and cardinalities of maximal cofinitary groups.…”

A Co-Analytic Cohen-Indestructible Maximal Cofinitary Group

“…We want to enlarge this group by σ * ∈ S ∞ , such that all of G, σ * is cofinitary. This can be done using a forcing invented by Zhang [22], which has proven extremely valuable in applications (see [2,23,8,24,14,7,13]). …”

“…the least size of a mcg, to other cardinal invariants of the continuum; see e.g. [22,23,8,2,6]. Analogous questions about permutation groups on κ, where κ is an uncountable cardinal, have also been studied; see e.g.…”

There is a Van Douwen MAD family

“…We will then show that we can, in fact, get a suitably "wide" slalom which is eventually disjoint from all slaloms in F (Lemma 2.6). Lemma 2.4 was independently discovered and used by Brendle, Spinas and Zhang [5].…”