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SUMMARY:Harder's reduction theory for S-arithmetic groups  over global fun
 ction fields - Ralf Gramlich (Birmingham)
DTSTART:20100519T153000Z
DTEND:20100519T163000Z
UID:TALK24326@talks.cam.ac.uk
CONTACT:Jan Saxl
DESCRIPTION:Let G be a reductive algebraic group over a global function fi
 eld K with\na residue field of order q\, let A_K be the ring of adeles ove
 r K\, and let P \nbe a maximal K-parabolic subgroup of G. Harder defines t
 he distance of a \nmaximal compact subgroup C of G(A_K) from P as the volu
 me of the\nintersection \nof C and P(A_K) with respect to the Tamagawa mea
 sure.\n\nFor each finite non-empty set S of places of K\, the logarithm of
  this \ndistance function to the basis q yields a Busemann function on the
  product\nof the affine buildings of G(K_s)\, where s runs through S. \n\n
 The family of all these Busemann functions\, as P varies through the maxim
 al\nK-parabolic subgroups of G\, allows one to define a Morse function on 
 this\nproduct of affine buildings\, which can be used to derive the finite
 ness \nproperties of the S-arithmetic subgroup of G(K).\n\nIn my talk I wi
 ll state Harder's fundamental theorem of reduction theory\,\nderive the Bu
 semann functions and the Morse function\, and explain how this\nsetup can 
 be used in order to establish finiteness properties of S-arithmetic groups
 .\n
LOCATION:MR12
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