[Paper Review] Groups of small homological dimension and the Atiyah Conjecture

Peter H. Kropholler, Peter A. Linnell+1 more|ArXiv.org|Jan 23, 2004

Homotopy and Cohomology in Algebraic Topology10 references7 citations

TL;DR

This paper establishes that a group of homological dimension ≤1 satisfying the Atiyah Conjecture about L²-Betti numbers must be locally free, providing a converse to the known result that locally free groups have homological dimension ≤1. It further shows that finitely generated elementary amenable groups of cohomological dimension ≤2 admit finite 2-dimensional models for BG and have 2-dimensional resolutions of the trivial ℤG-module ℤ.

ABSTRACT

A group G has homological dimension less or equal to 1 if it is locally free. We prove the converse provided that G satisfies the Atiyah Conjecture about L^2-Betti numbers. We also show that a finitely generated elementary amenable group G of cohomological dimension less or equal to 2 possesses a finite 2-dimensional model for BG and in particular that G is finitely presented and the trivial ZG-module Z has a 2-dimensional resolution by finitely generated free ZG-modules.

Motivation & Objective

• To establish a converse to the known fact that locally free groups have homological dimension ≤1, under the assumption of the Atiyah Conjecture.
• To characterize finitely generated elementary amenable groups of cohomological dimension ≤2 via their group presentations and geometric models.
• To show that such groups possess finite 2-dimensional classifying spaces BG and 2-dimensional resolutions of the trivial ℤG-module ℤ.
• To investigate the interplay between homological algebra, von Neumann dimension, and group-theoretic properties under the Atiyah Conjecture.

Proposed method

• Uses the Atiyah Conjecture on L²-Betti numbers to control the von Neumann dimension of modules over the group von Neumann algebra 𝒩(G).
• Applies Lemma 4 to show that projective modules with zero-dimensional quotient under tensoring with 𝒩(G) are finitely generated.
• Employs Lemma 5 to extract finitely generated submodules with zero-dimensional quotient in modules of finite von Neumann dimension.
• Leverages Stallings' result that groups of cohomological dimension 1 are free, under the FP condition.
• Applies Gildenhuys' theorem on one-relator presentations for solvable groups of cohomological dimension 2.
• Uses the fact that homology commutes with colimits to extend results from finitely generated subgroups to the whole group.

Experimental results

Research questions

• RQ1 Under what conditions does a group of homological dimension ≤1 necessarily have to be locally free?
• RQ2 Can the Atiyah Conjecture be used to characterize the group-theoretic structure of groups with low homological dimension?
• RQ3 What is the geometric and algebraic structure of finitely generated elementary amenable groups of cohomological dimension ≤2?
• RQ4 Do such groups admit finite 2-dimensional models for BG, and what does this imply about their resolutions?
• RQ5 How does the von Neumann dimension of modules over 𝒩(G) constrain the finite generation of projective modules?

Key findings

• A group of homological dimension ≤1 satisfying the Atiyah Conjecture must be locally free, providing a converse to the standard fact that locally free groups have homological dimension ≤1.
• For a finitely generated elementary amenable group of cohomological dimension ≤2, there exists a finite 2-dimensional model for BG, implying the group is finitely presented.
• The trivial ℤG-module ℤ admits a 2-dimensional resolution by finitely generated free ℤG-modules, a strong finiteness condition.
• Such groups are either non-cyclic subgroups of ℚ (if countable but not finitely generated) or admit a presentation ⟨x,y | yxy⁻¹ = xⁿ⟩ for some n ∈ ℤ.
• The proof relies on the Atiyah Conjecture to ensure that projective modules over 𝒩(H) with finite von Neumann dimension are finitely generated, enabling the use of Stallings' theorem.
• Elementary amenable groups of homological dimension ≤2 are metabelian, and their structure is constrained by the Hirsch length and the action of G/N on the Fitting subgroup.