Abstract
We apply a construction developed in a previous paper by the authors in order to obtain a formula which enables us to compute ell ^2-Betti numbers coming from a family of group algebras representable as crossed product algebras. As an application, we obtain a whole family of irrational ell ^2-Betti numbers arising from the lamplighter group algebra {mathbb Q}[{mathbb Z}_2 wr {mathbb Z}]. This procedure is constructive, in the sense that one has an explicit description of the elements realizing such irrational numbers. This extends the work made by Grabowski, who first computed irrational ell ^2-Betti numbers from the algebras {mathbb Q}[{mathbb Z}_n wr {mathbb Z}], where n ge 2 is a natural number. We also apply the techniques developed to the generalized odometer algebra {mathcal {O}}({overline{n}}), where {overline{n}} is a supernatural number. We compute its *-regular closure, and this allows us to fully characterize the set of {mathcal {O}}({overline{n}})-Betti numbers.
Highlights
In [5], Atiyah extended the Atiyah–Singer index theorem [6] to the non-compact setting and more concretely to the arena of non-compact manifolds M equipped with a free cocompact action of a discrete group G M
In his paper [5], Atiyah observed that, in case G is a finite group, the 2-Betti numbers coincide, modulo rescaling by |G|, with the previously known Betti numbers. In this situation, they are rational numbers, and Atiyah posed the following question, which is nowadays commonly known as the Atiyah Conjecture
Our main result (Theorem 4.5) provides a vast family of irrational, and even transcendental, 2-Betti numbers arising from the lamplighter group. We explore another method to find 2-Betti numbers arising from the lamplighter group, using the algebra of non-commutative rational series
Summary
In [5], Atiyah (in collaboration with Singer) extended the Atiyah–Singer index theorem [6] to the non-compact setting and more concretely to the arena of non-compact manifolds M equipped with a free cocompact action of a discrete group G M (i.e., with M/G compact). Turning back to the lamplighter group algebra K Γ , Graboswki has shown in a recent paper [18] the existence of irrational ( transcendental) 2-Betti numbers arising from Γ , exhibiting a concrete example in [18, Theorem 2]. It is not possible to realize the odometer algebra as a group algebra, this example is interesting in its own right because we are able to fully determine the structure of its ∗-regular closure RO (see Theorem 5.4), giving a complete description of the set of O-Betti numbers (Theorem 5.6). 5, the generalized odometer algebra O(n) in great detail, and we are able to completely determine the algebraic structure of its ∗-regular closure (Theorem 5.4) We use this characterization in Theorem 5.6, where we explicitly compute the whole set of O(n)-Betti numbers
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