Weakly surjunctive groups and symbolic group varieties

TL;DR

This paper introduces new classes of groups called weakly surjunctive and linearly surjunctive, explores their properties, and proves a reversibility theorem for injective endomorphisms of symbolic group varieties, providing new insights into Kaplansky's conjecture.

Contribution

It defines and studies weakly surjunctive and linearly surjunctive groups, extending the class of surjunctive groups, and proves a key reversibility theorem in this context.

Findings

Reversibility and invertibility theorem for injective endomorphisms

Inclusion of all sofic and surjunctive groups in the new classes

New evidence related to Kaplansky's stable finiteness conjecture

Abstract

In this paper, we introduce the classes of weakly surjunctive and linearly surjunctive groups which include all sofic groups and more generally all surjunctive groups. We investigate various properties of such groups and establish in particular a reversibility and invertibility theorem for injective endomorphisms of symbolic group varieties over weakly surjunctive group universes with algebraic group alphabets in arbitrary characteristic. We also obtain novel evidence related to Kaplansky's stable finiteness conjecture.