Closed manifolds with transcendentalL²-Betti numbers:

Abstract

In this paper, we show how to construct examples of closed manifolds with explicitly computed irrational, even transcendental L 2 Betti numbers, defined via the universal covering.We show that every non-negative real number shows up as an L 2 -Betti number of some covering of a compact manifold, and that many computable real numbers appear as an L 2 -Betti number of a universal covering of a compact manifold (with a precise meaning of computable given below).In algebraic terms, for many given computable real numbers (in particular for many transcendental numbers) we show how to construct a finitely presented group and an element in the integral group ring such that the L 2 -dimension of the kernel is the given number. We follow the method pioneered by Austin [2] but refine it to get explicit calculations which make the above statements possible.

Mean dimension, mean rank, and von Neumann–Lück rank

“…for any x, y ∈ X. Thus for any ε > max j>k λ j diam( [3,9,12,13,15,25,40] Coornaert and Krieger showed that if a countable amenable group Γ has subgroups of arbitrary large finite index, then for any r ∈ [0, +∞] there is a continuous action of Γ on some compact metrizable space X with mdim(X) = r [8]. Thus it is somehow surprising that the value of the mean dimension of algebraic actions of some amenable groups is rather restricted, as the following consequence of Theorems 1.1 and 8.3 shows: (see also [7] for some related discussion) We shall answer Question 9.1 for algebraic actions of the groups in Theorem 8.3.…”

Rational group ring elements with kernels having irrational dimension

“…Since a version of the current paper first appeared, its methods have been enhanced by Pichot, Schick andŻuk [16] and Grabowski [8] to obtain several further results. On the one hand, both sets of authors show that examples answering Atiyah's question may be found among amenable groups, and with kernel dimensions equal to any chosen element of [0, 1].…”

“…Remark. Since a version of the present paper first appeared online [2], studies by Pichot, Schick andŻuk [16] and Grabowski [8] have used a similar underlying construction to produce a range of related examples. By incorporating several non-trivial new ideas, those examples can be made simpler and more explicit than in the present paper, and can be arranged to have various additional properties such as amenability and finite presentation.…”

On computing homology gradients over finite fields

“…Only recently, motivated in part by the approach of [GŻ01] and [DS02], Austin [Aus13] showed that there exists a normal covering of a finite CW-complex with at least one irrational l 2 -Betti number. Several improvements followed shortly afterwards ( [Gra14] and [PSŻ15], [LW13], [Gra10]).…”

“…The reason why we cannot translate these results to an arbitrary field k is that their proofs employ the Higmann embedding theorem. As such, the examples in [Gra14] and [PSŻ15] have deck transformation groups which are not amenable in general, and consequently their k-homology gradients are not known to be well-defined.…”