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SUMMARY:Bianchi orbifolds and their torsion subcomplexes
DTSTART;VALUE=DATE-TIME:20150918T090000Z
DTEND;VALUE=DATE-TIME:20150918T100000Z
DTSTAMP;VALUE=DATE-TIME:20210413T094542Z
UID:indico-event-851@indico.math.cnrs.fr
DESCRIPTION:This talk starts with the Bianchi orbifolds\, which arise from
the action of the Bianchi groups (SL2 matrix groups over imaginary quadra
tic integers) on hyperbolic 3-space (their associated symmetric space). St
udying these orbifolds\, one can obtain : \n- dimensions of spaces of Bian
chi modular forms\; \n- Algebraic K-theory of rings of imaginary quadratic
integers\; \n- Group (co)homology and equivariant K-homology of the Bianc
hi groups\; \n- Chen-Ruan orbifold cohomology of the Bianchi orbifolds. Th
e latter cohomology ring is conjectured to be isomorphic to the cohomology
ring of a crepant resolution for the orbifold\, which makes it interestin
g for string theory. \nWhat makes Bianchi modular forms interesting\, is t
hat there are deep number-theoretical reasons for expecting that the Taniy
ama-Shimura correspondence can extend to the cuspidal Bianchi modular form
s\, attaching Abelian varieties to them. For the dimension calculations\,
use is made of the speaker's constructive answer to a question of Jean-Pie
rre Serre on the Borel-Serre compactification of the Bianchi orbifolds\, w
hich had been open for 40 years. This has so far permitted heavy machine c
alculations joint with M. Haluk Sengun\, locating several of the very rare
instances of Bianchi modular forms in a large array of dicriminants and w
eights\, such that these forms are not lifts of classical modular forms. \
nComplementary to this\, studies of the torsion part of the cohomology of
the Bianchi groups have given rise to a new technique (called Torsion Subc
omplex Reduction\; some procedures of the technique had beforehand been us
ed as ad hoc tricks by Soulé\, Mislin and Henn) for computing the Farrell
-Tate cohomology of discrete groups acting on suitable cell complexes. Thi
s technique has not only already yielded general formulae for the cohomolo
gy of the tetrahedral Coxeter groups as well as\, above the virtual cohomo
logical dimension\, of the Bianchi groups (and at odd torsion\, more gener
ally of SL2 groups over arbitrary number fields\, in joint work with Matth
ias Wendt)\, it also very recently has allowed Wendt to refine the Quillen
conjecture. \nThis talk will further discuss the adaptation of this techn
ique to Bredon homology computations\, yielding the equivariant K-homology
of the Bianchi groups. The Baum-Connes conjecture\, which has been proved
for large classes of groups\, constructs an (iso)morphism from the equiva
riant K-homology of a group to the K-theory of its reduced C*-algebra. For
the Bianchi groups\, this yields the isomorphism type of the mentioned op
erator K-theory\, which would be very hard to compute directly.\n\nhttps:/
/indico.math.cnrs.fr/event/851/
LOCATION:IHES Amphithéâtre Léon Motchane
URL:https://indico.math.cnrs.fr/event/851/
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