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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the withdrawn paper arXiv:math/0607470.Comment: 4 page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
Approximating L2-invariants, and the Atiyah conjecture
Let G be a torsion free discrete group and let \bar{Q} denote the field of
algebraic numbers in C. We prove that \bar{Q}[G] fulfills the Atiyah conjecture
if G lies in a certain class of groups D, which contains in particular all
groups which are residually torsion free elementary amenable or which are
residually free. This result implies that there are no non-trivial
zero-divisors in C[G]. The statement relies on new approximation results for
L2-Betti numbers over \bar{Q}[G], which are the core of the work done in this
paper.
Another set of results in the paper is concerned with certain number
theoretic properties of eigenvalues for the combinatorial Laplacian on
L2-cochains on any normal covering space of a finite CW complex.
We establish the absence of eigenvalues that are transcendental numbers,
whenever the covering transformation group is either amenable or in the Linnell
class \mathcal{C}. We also establish the absence of eigenvalues that are
Liouville transcendental numbers whenever the covering transformation group is
either residually finite or more generally in a certain large bootstrap class
\mathcal{G}. Please take the errata to Schick: "L2-determinant class and
approximation of L2-Betti numbers" into account, which are added at the end of
the file, rectifying some unproved statements about "amenable extension". As a
consequence, throughout, amenable extensions should be extensions with normal
subgroups.Comment: AMS-LaTeX2e, 33 pages; improved presentation, new and detailed proof
about absence of trancendental eigenvalues; v3: added errata to
"L2-determinant class and approximation of L2-Betti numbers", requires to
restrict to slightly weaker statement
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