Burghelea conjecture and asymptotic dimension of groups

TL;DR

This paper explores the Burghelea conjecture's relation to group properties, proves conjectures for many groups, and constructs counter-examples, including a finitely presentable one based on Thompson's group F.

Contribution

It introduces two conjectures linking asymptotic dimension to the Burghelea conjecture and constructs the first finitely presentable counter-example.

Findings

Proved conjectures for many classes of groups.

Constructed a finitely presentable counter-example based on Thompson's group F.

Identified limitations of the Burghelea conjecture with counter-examples.

Abstract

We review the Burghelea conjecture, which constitutes a full computation of the periodic cyclic homology of complex group rings, and its relation to the algebraic Baum-Connes conjecture. The Burghelea conjecture implies the Bass conjecture. We state two conjectures about groups of finite asymptotic dimension, which together imply the Burghelea conjecture for such groups. We prove both conjectures for many classes of groups. It is known that the Burghelea conjecture does not hold for all groups, although no finitely presentable counter-example was known. We construct a finitely presentable (even type $F_\infty$) counter-example based on Thompson's group F. We construct as well a finitely generated counter-example with finite decomposition complexity.