The strong Atiyah and L\"uck approximation conjectures for one-relator groups

TL;DR

This paper proves that the strong Atiyah and L"uck approximation conjectures hold for one-relator groups, along with several other conjectures, by establishing properties of locally indicable groups and their group algebras.

Contribution

It demonstrates that one-relator groups satisfy key conjectures in group theory and operator algebras, extending the class of groups for which these conjectures are verified.

Findings

Strong Atiyah conjecture holds for one-relator groups.

L"uck approximation conjecture holds for locally indicable groups.

Group algebra over characteristic zero fields embeds into a division algebra.

Abstract

It is shown that the strong Atiyah conjecture and the L\"uck approximation conjecture in the space of marked groups hold for locally indicable groups. In particular, this implies that one-relator groups satisfy both conjectures. We also show that the center conjecture, the independence conjecture and the strong eigenvalue conjecture hold for these groups. As a byproduct we prove that the group algebra of a locally indicable group over a field of characteristic zero has a Hughes-free epic division algebra and, in particular, it is embedded in a division algebra.