Petyarra and Moffatt: 'Looking from the sky'

Moffatt’s Up in the Sky series draws attention to the relation between sky and earth, through the content and camera angles of the images. Similarly, Kathleen Petyarre’s Central Desert acrylic dot painting evokes this relation representing country and Dreaming from a celestial perspective—as she says ‘looking from the sky’. Yet here any association between these artists seems to end with the urban artist refusing to engage Aboriginal tradition and the desert artist focused on Dreaming, country and heritage. However, a further connection between these disparate works may also be discerned as each, in differing ways, transforms our conventional perceptions of space and time. Reading these images in relation to Walter Benjamin’s concepts of the auratic and of messianic time, I suggest that each restructures dimension and duration putting in question the (post)modern calibrations of our space/time experience. This paper stages an engagement between these artists’ works and Benjamin’s concepts exploring the variations and modifications of the spatial and the temporal that hybrid cross-cultural exchanges require and facilitate

The space of left orders of a group is either finite or uncountable

Let G be a group and let O_G denote the set of left orderings on G. Then O_G can be topologized in a natural way, and we shall study this topology to show that O_G can never be countably infinite. This paper retrieves correct parts of the withdrawn paper arXiv:math/0607470.Comment: 4 page

Right orderable residually finite p-groups and a Kourovka notebook problem

A. H. Rhemtulla proved that if a group is a residually finite p-group for infinitely many primes p, then it is two-sided orderable. In problem 10.30 of the Kourovka notebook 14th. edition, N. Ya. Medvedev asked if there is a non-right-orderable group which is a residually finite p-group for at least two different primes p. Using a result of Dave Witte, we will show that many subgroups of finite index in GL_3(Z) give examples of such groups. On the other hand we will show that no such example can exist among solvable by finite groups.Comment: 2 pages, to appear in J. Algebr

A rationality criterion for unbounded operators

Let G be a group, let U(G) denote the set of unbounded operators on L^2(G) which are affiliated to the group von Neumann algebra W(G) of G, and let D(G) denote the division closure of CG in U(G). Thus D(G) is the smallest subring of U(G) containing CG which is closed under taking inverses. If G is a free group then D(G) is a division ring, and in this case we shall give a criterion for an element of U(G) to be in D(G). This extends a result of Duchamp and Reutenauer, which was concerned with proving a conjecture of Connes.Comment: 7 pages, to appear in the Comptes Rendu

Finite group extensions and the Atiyah conjecture

The Atiyah conjecture for a discrete group G states that the $L^2$-Betti numbers of a finite CW-complex with fundamental group G are integers if G is torsion-free and are rational with denominators determined by the finite subgroups of G in general. Here we establish conditions under which the Atiyah conjecture for a group G implies the Atiyah conjecture for every finite extension of G. The most important requirement is that the cohomology $H^*(G,\mathbb{Z}/p)$ is isomorphic to the cohomology of the p-adic completion of G for every prime p. An additional assumption is necessary, e.g. that the quotients of the lower central series or of the derived series are torsion-free. We prove that these conditions are fulfilled for a class of groups which contains Artin's pure braid groups, free groups, surfaces groups, certain link groups and one-relator groups. Therefore every finite, in fact every elementary amenable extension of these groups satisfies the Atiyah conjecture. In the course of the proof we prove that if these extensions are torsion-free, then they have plenty of non-trivial torsion-free quotients which are virtually nilpotent. All of this applies in particular to Artin's full braid group, therefore answering question B6 on http://www.grouptheory.info . Our methods also apply to the Baum-Connes conjecture. This is discussed in arXiv:math/0209165 "Finite group extensions and the Baum-Connes conjecture", where the Baum-Connes conjecture is proved e.g. for the full braid group.Comment: 54 pages, typos and small mistakes corrected, final version to appear in Journal of the AM