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SUMMARY:On l^2-Betti numbers and their analogues in positive characteristi
 c - Andre Jaikin (Madrid)
DTSTART:20170217T134500Z
DTEND:20170217T150000Z
UID:TALK71034@talks.cam.ac.uk
CONTACT:Maurice Chiodo
DESCRIPTION:Let G be a group\, K a field and A an nxm matrix over the grou
 p ring K[G]. Let G=G1>G2>G3... be a chain of normal subgroups of G of fini
 te index with trivial intersection. The multiplication on the right side b
 y A induces  linear maps\nΦ_i :   K[G/Gi]^n^  ->  K[G/Gi]^m^\nwith Φ_i(v
 1\,...\,vn) = (v1\,...\,vn)A\n\nWe are interested in  properties of the se
 quence $\\{\\frac{\\dim_K \\ker \\phi_i}{|G:G_i|}\\}$. In particular\, we 
 would like to answer the following questions:\n\n1. Is there the limit $ \
 \lim_{i\\to \\infty}\\frac{\\dim_K \\ker \\phi_i}{|G:G_i|}$?\n\n2. If the 
 limit exists\, how does it depend on the chain {G_i}?\n\n3. What is the ra
 nge of possible values for  $ \\lim_{i\\to \\infty}\\frac{\\dim_K \\ker \\
 phi_i}{|G:G_i|}$ for a given group G?\n\nIt turns out that  the answers on
  these questions are known for many groups G if K is a number field\, less
  known if K is an arbitrary field of characteristic 0 and almost unknown i
 f K is a field of positive characteristic.\n\nIn my talk I will give sever
 al motivations to consider these questions\, describe the known results an
 d present recent advances in the case where K has  characteristic 0.
LOCATION:CMS\, MR13
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