A groupoid approach to L\"uck's amenability conjecture

TL;DR

This paper establishes a new characterization of group amenability through the lens of dimension flatness in ring inclusions related to measure-preserving group actions, providing a solution to Lück's conjecture.

Contribution

It introduces a group-measure space framework to prove that amenability is equivalent to dimension flatness of specific ring inclusions, solving Lück's conjecture.

Findings

Amenability of a group is equivalent to dimension flatness of certain ring inclusions.

Provides a measure-theoretic approach to Lück's conjecture.

Connects group properties with operator algebraic structures.

Abstract

We prove that amenability of a discrete group is equivalent to dimension flatness of certain ring inclusions naturally associated with measure preserving actions of the group. This provides a group-measure space theoretic solution to a conjecture of L\"uck stating that amenability of a group is characterized by dimension flatness of the inclusion of its complex group algebra into the associated von Neumann algebra.