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Organizers |
Isometries of infinite-dimensional Riemannian manifolds
by
Christopher Atkin
Victoria University of Wellington
The group of self-isometries of a complete infinite-dimensional Riemannian manifold is a Lie group (in principle of infinite dimension).
Date received: November 2, 2007
Costs of equivalence relations and group actions
by
Anthony Dooley
UNSW, Sydney
Coauthors: V. Golodets
Much work has been done in studying amenable group actions, but until recently it has been difficult to handle non-amenable actions (or equivalence relations). A breakthrough was made with work of Levitt, Kechris and Gaboriau to define a new invariant, the cost.
Gaboriau showed how to use this invariant to distinguish between group actions of, for example, the free group on two generators and the free group on three generators.
Golodets and I use the theory of index cocycles of Feldman, Sutheraland and Zimmer, to calculate the cost of equivalence relations which are finite extensions. This enables us to resolve some conjectures of Gaboriau and also to show that many group actions cannot be isomorphic.
I will give an outline of the theory of costs and outline our main results.
Date received: October 30, 2007
Orbit inequivalent actions of non-amenable groups
by
Inessa Epstein
University of California - Los Angeles
Let G be a countable group acting in a Borel way on a standard probability space X. The orbits of this action give rise to an equivalence relation on X. We say two measure preserving actions of groups G and H on spaces X and Y, respectively, are orbit equivalent if there is a measure preserving bijection between conull subsets of X and Y identifying the orbits. We discuss a result that every non-amenable group admits continuum many orbit inequivalent free, measure preserving, ergodic actions.
Date received: October 29, 2007
Groups acting on Banach spaces
by
Stefano Ferri
Universidad de los Andes
Coauthors: Jorge Galindo Pastor
We shall present techniques to determine when a topological group can act as isometries on a "nice" Banach space (where "nice" could mean Hilbert, reflexive, Asplund...) and study in details the case of of groups which act on reflexive spaces.
Date received: October 30, 2007
On group algebras for non-locally compact groups
by
Hendrik Grundling
University of New South Wales
Coauthors: Karl-Hermann Neeb
We generalise group algebras to other algebraic objects with bounded Hilbert space representation theory - the generalised group algebras are called "host" algebras. The main property of a host algebra, is that its representation theory should be isomorphic (in the sense of the Gelfand-Raikov theorem) to a specified subset of representations of the algebraic object. The main motivation behind this, comes from the analysis of infinite dimensional Lie groups and other non-locally compact groups (some of which occur in physics).
In recent work on the topic we analyzed ordinary and multiplier (unitary) representations for non-locally compact Abelian groups, and found that host algebras need not exist, nor be unique if they do exist.
On the positive side, we constructed a host algebra for the multiplier representation theory associated to a fixed 2-cocycle of a non-locally compact Abelian group. I will sketch this construction. This has direct application to the canonical commutation relations of quantum fields.
Date received: October 31, 2007
The Structure of Connected Pro-Lie Groups
by
Sidney A. Morris
University of Ballarat
This talk is an introduction to connected pro-Lie groups and their structure as recently appeared in the book "The Lie Theory of Connected Pro-Lie Groups" by Karl Heinrich Hofmann and Sidney A. Morris and published by the European Mathematical Society.
Date received: October 15, 2007
A footnote to the property (FH)
by
Vladimir Pestov
Department of Mathematics and Statistics, University of Ottawa, Ontario, Canada
For second countable locally compact groups G (though not for more general classes of even discrete groups), Kazhdan's property (T) is equivalent to the following property known as (FH): every continuous action of G by affine isometries on a Hilbert space has a fixed point. Recently there has been some interest towards versions of this property stated for more general classes of Banach spaces, especially uniformly convex spaces, including Lp spaces (cf. a recent preprint by Bader, Furman, Gelander, and Monod, arXiv:math/0506361).
In particular, Haagerup and Przybyszewska have shown [arXiv:math/0606794] that every second countable locally compact non-compact group admits a continuous affine action by isometries without fixed points on a strictly convex (reflexive) Banach space.
One cannot hope to extend this result to non locally compact Polish groups, because, by force of a theorem by Megrelishvili [Semigroup Forum 63 (2001), 357-370] stating that every WAP function on the Polish group Homeo+[0, 1] is constant, this particular group admits no nontrivial continuous affine actions by isomeries on reflexive Banach spaces. Nevertheless, we observe that every topological group G that is not precompact admits a continuous affine actiion by isometries on a Banach space without fixed points. In fact, this property characterizes precompactness.
The proof uses a novel characterization by Uspenskij [arXiv:math/0004119] of precompact groups as those topological groups G in which every neighbourhood of the identity, U, admits a finite set F with FUF=G. Another component of the proof is the following observation of independent interest: every continuous action of a topological group G by isometries on a metric space X extends to an affine isometric action of G on a suitable Banach space containing X as a subspace and affinely spanned by it.
Date received: October 23, 2007
Generic representations of finitely generated groups.
by
Christian Rosendal
University of Illinois at Urbana-Champaign
For finitely generated groups G and ultrahomogeneous countable relational structures M we study the space Rep(G, M) of all representations of G by automorphisms on M equipped with the topology it inherits seen as a closed subset of Aut(M)G;. When G is the free group on n generators this space is just Aut(M)n, but is in general significantly more complicated. We prove that when G is finitely generated abelian and M the random structure of a finite relational language or the random ultrametric space of a countable distance set there is a generic point in Rep(G, M), i.e., there is a comeagre set of mutually conjugate representations of G on M. This is analogous to results of Hrushovski, Herwig, and Herwig–Lascar for the case G = Fn.
Date received: October 30, 2007
Oscillation stability for topological groups and Ramsey theory.
by
Lionel Nguyen Van Thé
University of Calgary, Canada
Coauthors: Jordi Lopez-Abad (Université Paris 7), Norbert Sauer (University of Calgary).
In 2003, Kechris Pestov and Todorcevic established several connections between dynamics of topological groups and combinatorics. Among the concepts that were then introduced stands the so-called 'oscillation stability for topological groups'. Very few results about this notion are currently known. One of the most important ones was obtained by Hjorth in 2006 and states that no non-trivial Polish group G is such that the (G, {e}) is oscillation stable. Another important example comes from the reformulation of the solution of the so-called distortion problem for l2 due to Odell and Schlumprecht in 1994 and states that if G is the surjective isometry group of the unit sphere of the Hilbert space l2 and Stx is the stabilizer of an element x in the sphere, then (G, Stx) is never oscillation stable. The purpose of the present talk is to show that the situation is quite different if the latter problem is considered when the unit Hilbert sphere is replaced by another remarkable Polish metric space: the Urysohn sphere.
Date received: October 25, 2007
Full Groups of Equivalence Relations
by
Todor Tsankov
California Institute of Technology
Coauthors: John Kittrell
We study full groups of countable, measure-preserving equivalence relations. By a classical theorem of Dye, those groups are complete invariants for the equivalence relations (up to a.e. isomorphism). We show that the (non-trivial) full groups are homeomorphic to Hilbert space and that homomorphisms from ergodic ones to arbitrary separable groups are continuous. We also find bounds for the minimal number of topological generators (elements generating a dense subgroup) of full groups allowing us to distinguish full groups of equivalence relations generated by free, ergodic actions of the free groups Fn and Fm if m and n are sufficiently far apart. We also show that an ergodic equivalence relation is generated by an action of a finitely generated group iff its full group is topologically finitely generated.
Date received: October 30, 2007
On finite groups in Stone-Cech compactifications
by
Yevhen Zelenyuk
School of Mathematics, University of the Witwatersrand, South Africa
The Stone-Cech compactification of an infinite discrete semigroup is an important object interesting both for its own sake and for its applications to combinatorial number theory and to topological dynamics. It is known that if the semigroup is cancellative, the Stone-Cech compactification contains large free groups. We shall discuss the question whether it contains any nontrivial finite group.
Date received: October 7, 2007