Generic torsion-free groups and Rubin actions

TL;DR

This work studies the extent to which generic countable torsion-free groups can act on topological spaces in Rubin-like ways. Using model-theoretic forcing via Banach–Mazur games, it forces algebraic disjointness and conjugacy properties in a comeager set of torsion-free groups, resulting in groups with rich internal structure but prohibiting Rubin or weakly Rubin actions. The authors show that, although generic groups cannot realize Rubin actions, their Rubin posets are nonetheless highly intricate, containing bi-infinite chains and infinite antichains built from cyclic subgroups, with embeddings of $ obreak \mathbb{Z}^2$ and rank-1 divisible components like $ obreak ext{$ ational$}$. This reveals a sharp separation between the existence of rich subgroup configurations and the obstruction to global Rubin actions, informing obstructions to group actions on compact manifolds and highlighting the nuanced landscape of generic torsion-free groups.

Abstract

We use model theoretic forcing to prove that a generic countable torsion-free group does not admit any nontrivial locally moving action on a Hausdorff topological space, and yet admits a rich Rubin poset.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Proposition 2.1: See Rubin1996rubin-shortKK2021book
• Lemma 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof : Proof of Lemma \ref{['lem:overgroup']}
• Lemma 3.4