On the Idempotent Conjecture for Sidki Doubles

TL;DR

This paper investigates the idempotent conjecture for Sidki doubles of torsion-free groups, providing partial results that advance understanding of the conjecture in this specific algebraic context.

Contribution

It offers the first partial verification of the idempotent conjecture for Sidki doubles of torsion-free groups, a novel extension in group ring theory.

Findings

Partial verification of the idempotent conjecture for Sidki doubles

Identification of conditions under which the conjecture holds for these doubles

Extension of the conjecture's study to a new class of groups

Abstract

Let $G$ be a group and $X(G)$ its Sidki Double. The idempotent conjecture says that there should be no non-trivial idempotent in the complex group ring of a torsion-free group. We investigate this conjecture for the Sidki double of a torsion-free group, and obtain a partial result.