138 research outputs found
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the withdrawn paper arXiv:math/0607470.Comment: 4 page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 -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
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
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