Topological Hochschild homology and the Bass trace conjecture

TL;DR

This paper applies topological Hochschild homology to analyze the Bass trace conjecture, revealing new restrictions on groups satisfying the conjecture and confirming it for certain groups with specific subgroup properties.

Contribution

It demonstrates that the factorization of the Hattori-Stallings rank through the cyclotomic trace imposes new restrictions on groups satisfying the Bass trace conjecture.

Findings

The factorization leads to Linnell's restriction on such groups.

The conjecture holds for groups where rational subgroup images are nontrivial and central in some quotient.

Provides new insights into the structure of groups satisfying the conjecture.

Abstract

We use the methods of topological Hochschild homology to shed new light on the groups satisfying the Bass trace conjecture. We show that the factorization of the Hattori-Stallings rank map through the Bokstedt-Hsiang-Madsen cyclotomic trace map leads to Linnell's restriction on such groups. As a new consequence of this restriction, we show that the conjecture holds for any group G with the property that every subgroup that is isomorphic to the additive group of rational numbers has nontrivial and central image in some quotient of G.