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JOINT TAU-TECHNION SEMINAR ON GEOMETRY & DYNAMICS
 <http://www.math.tau.ac.il/~ostrover/Seminars/GD/Seminar_GD.html>
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TECHNION
 
DATE: Wednesday, March 30, 2011, 13:00
 
SPEAKER: Egor Shelukhin (Tel-Aviv University)
 
TITLE: Moment maps and quasimorphisms
 
LOCATION: Amado bldg., room 814, Technion
 
ABSTRACT: It has long been known (and is due to many authors) that
whenever a group G acts on a Hermitian symmetric space of non-compact type
by Kahler isometries, one can construct a bounded two-cocycle on G by
integrating the Kahler form over geodesic triangles. Similarly, Reznikov
has constructed bounded two-cocycles on groups of symplectomorphisms using
their action on the space of compatible almost complex structures. We show
that if the action under discussion has an equivariant moment map, such a
cocycle has a primitive - a quasimorphism on the universal cover of the
group. This holds in the finite-dimensional case - that is for Hermitian
Lie groups - giving a reinterpretation of the Guichardet-Wigner
quasimorphisms, and for the infinite-dimensional groups of Hamiltonian
diffeomorphisms of any finite volume symplectic manifold, generalizing
several previous constructions due to Barge-Ghys, Entov and Py. The moment
map construction in the second case is due to Donaldson and Fujiki (for
the integrable structures). We also compute the restriction of the
quasimorphism to the fundamental group and determine its local type. Our
construction involves a generalization of Weinstein's Action homomorphism
and is related to the Barge-Ghys construction for discrete subgroups of
PSL(2,R).
 
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Technion Math. Net (TECHMATH)
Editor: Michael Cwikel   <techm@math.technion.ac.il> 
Announcement from: Michael Entov   <entov@math.technion.ac.il>