Traces on group C*-algebras, sofic groups and L\"uck's conjecture

TL;DR

This paper provides an alternative proof of L"uck's determinant conjecture for sofic groups using traces on group C*-algebras, and discusses related issues in $L^2$-Betti numbers and Atiyah's problem.

Contribution

It introduces a new proof technique for L"uck's conjecture for sofic groups based on traces on group C*-algebras, offering fresh insights.

Findings

Proof of L"uck's determinant conjecture for sofic groups

Connection between traces on group C*-algebras and L"uck's conjecture

Discussion on Atiyah's problem and $L^2$-Betti numbers

Abstract

We give an alternate proof of a Theorem of Elek and Szabo establishing L\"uck's determinant conjecture for sofic groups. Our proof is based on traces on group C*-algebras. We briefly discuss the relation with Atiyah's problem on the integrality of $L^2$-Betti numbers.