Groups satisfying Kaplansky's stable finiteness conjecture

TL;DR

This paper proves that certain residually finite-by-sofic groups satisfy Kaplansky's conjectures, providing new examples of non-LEA groups where soficity remains unresolved, expanding understanding of group algebra properties.

Contribution

It establishes that finitely generated residually finite-by-sofic groups satisfy Kaplansky's stable finiteness conjectures, introducing new non-LEA group examples with unresolved soficity.

Findings

Residually finite-by-sofic groups satisfy Kaplansky's conjectures

Provides new examples of non-LEA groups with unresolved soficity

Includes notable examples like Deligne's group and specific amalgamated free products

Abstract

We prove that every {finitely generated residually finite}-by-sofic group satisfies Kaplansky's direct and stable finiteness conjectures with respect to all noetherian rings. We use this result to provide countably many new examples of finitely presented non-LEA groups, for which soficity is still undecided, satisfying these two conjectures. Deligne's famous example of a non residually finite group is among our examples, along with the families of amalgamated free products SL_n(Z[1/p])*_{F_r}SL_n(Z[1/p]) and HNN extensions SL_n(Z[1/p])*_{F_r}, where p>2 is a prime, n>2 and F_r is a free group of rank r, for all r>1.