[Paper Review] On the Atiyah problem for the lamplighter groups

Łukasz Grabowski|arXiv (Cornell University)|Sep 1, 2010

Mathematics and Applications7 references11 citations

TL;DR

This paper disproves a conjecture by Lott and Lück by demonstrating that the Novikov-Shubin invariant of an element in the integral group ring of the lamplighter group Z₂ ≀ ℤ can be irrational. It further shows that every positive real number arises as such an invariant, and that the ℓ²-Betti numbers of matrices over the integral group ring of Zₚ ≀ ℤ (p > 1) can also be irrational, establishing these groups as the simplest known to produce irrational ℓ²-Betti numbers.

ABSTRACT

We show that the Novikov-Shubin invariant of an element of the integral group ring of the lamplighter group Z_2 \wr Z can be irrational. This disproves a conjecture of Lott and Lueck. Furthermore we show that every positive real number is equal to the Novikov-Shubin invariant of some element of the real group ring of Z_2 \wr Z. Finally we show that the l2-Betti number of a matrix over the integral group ring of the group Z_p \wr Z, p>1, can be irrational, and so the groups Z_p \wr Z become the simplest known groups which give rise to irrational l2-Betti numbers.

Motivation & Objective

• To investigate the nature of Novikov-Shubin invariants in the context of group rings of lamplighter groups.
• To test the validity of a conjecture by Lott and Lück asserting that Novikov-Shubin invariants must be rational for elements in the integral group ring of Z₂ ≀ ℤ.
• To determine the full range of possible values for Novikov-Shubin invariants in the real group ring of Z₂ ≀ ℤ.
• To examine whether ℓ²-Betti numbers of matrices over the integral group ring of Zₚ ≀ ℤ can be irrational for p > 1.
• To identify the simplest groups that yield irrational ℓ²-Betti numbers, establishing Zₚ ≀ ℤ as such groups.

Proposed method

• Construction of specific elements in the integral group ring of Z₂ ≀ ℤ to analyze their associated Novikov-Shubin invariants.
• Use of spectral theory and analytic properties of group operators to evaluate the growth rate of the spectral density function near zero.
• Application of techniques from L²-cohomology and operator algebras to study ℓ²-Betti numbers of matrices over group rings.
• Generalization of results from Z₂ ≀ ℤ to Zₚ ≀ ℤ for p > 1 to demonstrate the existence of irrational ℓ²-Betti numbers.
• Proof that every positive real number is realized as a Novikov-Shubin invariant via a constructive argument in the real group ring.
• Leveraging the structure of the lamplighter group as a wreath product to exploit its self-similar and amenable properties in spectral computations.

Experimental results

Research questions

• RQ1 Can the Novikov-Shubin invariant of an element in the integral group ring of Z₂ ≀ ℤ be irrational, contradicting the conjecture of Lott and Lück?
• RQ2 Is every positive real number achievable as the Novikov-Shubin invariant of some element in the real group ring of Z₂ ≀ ℤ?
• RQ3 Do ℓ²-Betti numbers of matrices over the integral group ring of Zₚ ≀ ℤ (p > 1) admit irrational values?
• RQ4 What is the minimal complexity of a group that can produce irrational ℓ²-Betti numbers?
• RQ5 Are the lamplighter groups Zₚ ≀ ℤ the simplest known groups to exhibit irrational ℓ²-Betti numbers?

Key findings

• The Novikov-Shubin invariant of an element in the integral group ring of Z₂ ≀ ℤ can be irrational, thereby disproving the conjecture of Lott and Lück.
• Every positive real number is realized as the Novikov-Shubin invariant of some element in the real group ring of Z₂ ≀ ℤ.
• The ℓ²-Betti number of a matrix over the integral group ring of Zₚ ≀ ℤ (p > 1) can be irrational.
• The groups Zₚ ≀ ℤ are the simplest known groups that give rise to irrational ℓ²-Betti numbers.
• The results establish that irrationality in ℓ²-invariants is not restricted to highly complex groups, but occurs already in these relatively simple wreath products.
• The findings extend the understanding of spectral invariants in amenable and self-similar groups, particularly in the context of L²-theory and group ring representations.