Connecting the space of marked groups and the space of group operations

TL;DR

This paper bridges two canonical encodings of countable groups—the space of group operations $\mathcal{G}$ and the space of marked groups $\mathcal{M}$—by constructing a comeager domain $\mathcal{D}\subseteq\mathcal{M}$ and a surjective open map $f:\mathcal{D}\to\mathcal{G}$ with $F_\infty/N\cong\widetilde{f(N)}$ for all $N\in\mathcal{D}$. This yields an equivalence of generic properties between the two spaces and enables translating a wide array of results from $\mathcal{G}$ to $\mathcal{M}$, including new, shorter arguments for the generic behavior of compact metrizable abelian groups. The work also clarifies limitations of the natural map $\Phi:\mathcal{G}\to\mathcal{M}$ by showing its range is nowhere dense, hence it cannot transfer genericity directly. The framework unifies and extends known 0-1 laws, meagerness of isomorphism classes, and comeager isomorphism types across subspaces, with concrete consequences for abelian and compact metrizable abelian group spaces via Pontryagin duality.

Abstract

We establish a connection between two well-studied spaces of countable groups: the space of group operations and the space of marked groups. This connection shows that the two spaces are equivalent in terms of generic properties in the sense of Baire category, which allows us to translate several results from the former setting to the latter. As an application, we give new, shorter proofs of two theorems that concern the generic behavior of compact metrizable abelian groups.

Figures (1)

• Figure 1:

Theorems & Definitions (44)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1
• Theorem 2.2: Baire Category Theorem
• Proposition 2.3
• Proposition 2.4
• proof
• Remark 2.5