Free groups are $L^2$-subgroup rigid

TL;DR

This work defines $L^2$-subgroup rigidity and proves that free groups are $L^2$-subgroup rigid, establishing a robust link between subgroup closures and algebraic invariants. The authors develop a comprehensive algebraic framework using left ideals ${}^G I_{k[H]}$, universal division rings, and $L^2$-torsion to define $L^2$-closures of finitely generated subgroups. They prove that, for a finitely generated subgroup $H$ of a free group $F$, the $L^2$-closure exists and yields equivalences among compressed, inert, strictly inert, and $L^2$-independent behavior, unifying several prior conjectures. Moreover, they show that the $L^2$-closure can be characterized via torsion computations in $ ext{Q}[F]$ and $ ext{F}_2[F]$, and they derive structural consequences for the lattice of finitely generated subgroups, including the notion of a Crit set and its stability under intersections and joins. The results provide a powerful algebraic approach to subgroup theory in free groups and suggest pathways to extending these rigidity phenomena to broader classes of groups, with implications for decidability and primitivity-related invariants.

Abstract

In this paper, we introduce the notion of $L^2$-subgroup rigid groups and demonstrate that free groups are $L^2$-subgroup rigid. As a consequence, we establish the equivalence between compressibility, inertness, strong inertness, and $L^2$-independence for a finitely generated subgroup of a free group, confirming a conjecture by Dicks and Ventura as well as the one by Antolin and Jaikin-Zapirain.

Theorems & Definitions (52)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Corollary 1.4
• Corollary 1.5
• Lemma 2.1
• proof
• Lemma 2.2
• Proposition 2.3
• Lemma 2.4