On minimal non-sofic and $\omega$-non-sofic groups

TL;DR

This paper explores the structural properties of hypothetical non-sofic groups, introducing classes like minimal non-sofic and $oldsymbol{oldsymbol{oldsymbol{oldsymbol{ ext{omega}}}}}$-non-sofic groups, and establishing their key characteristics and implications.

Contribution

It defines and analyzes minimal non-sofic and $oldsymbol{oldsymbol{oldsymbol{oldsymbol{ ext{omega}}}}}$-non-sofic groups, revealing their structural restrictions and implications for the existence of non-sofic groups.

Findings

Minimal non-sofic groups have a central finitely generated residually finite maximal normal subgroup.

Locally graded non-sofic groups are necessarily $oldsymbol{oldsymbol{oldsymbol{oldsymbol{ ext{omega}}}}}$-non-sofic.

Existence of a non-sofic group implies existence of a countable existentially closed non-sofic group with a densely ordered chain of centralizers.

Abstract

We investigate structural properties of non-sofic groups, assuming that such groups exist. We introduce and study two classes: minimal non-sofic groups and $\omega$-non-sofic groups. For minimal non-sofic groups, we establish strong structural restrictions. In particular, we show that if $G$ is a minimal non-sofic group and $M$ is a finitely generated residually finite maximal normal subgroup of $G$, then $M$ is central and $G$ is a perfect central extension of a finitely generated non-amenable simple group. On the other hand, we show that locally graded non-sofic groups are necessarily $\omega$-non-sofic. More precisely, such groups contain finitely generated non-sofic subgroups admitting strictly decreasing chains of finitely generated normal subgroups whose intersection is nontrivial and lies in the profinite residual. Finally, using results on existentially closed groups, we prove that the existence of a non-sofic group implies the existence of a countable existentially closed non-sofic group whose centralizers form a densely ordered chain of non-sofic subgroups of order type $(\mathbb{Q},\leq)$. In particular, we show that if a non-sofic group exists, then the class of $\omega$-non-sofic groups is non-empty. Moreover, we prove that the existence of a non-sofic group implies the existence of a non-sofic group of unbounded exponent.