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CATEGORIES:Groups St Andrews 2017
SUMMARY:On ℓ^2-Betti numbers and their analogues in positi
ve characteristic - Andrei Jaikin-Zapirain\, Unive
rsidad Autónoma de Madrid
DTSTART:20170812T083000Z
DTEND:20170812T093000Z
UID:TALK2733AT
URL:/talk/index/2733
DESCRIPTION:Let _G_ be a group\, _K_ a field and _A_ a _n_ by
_m_ matrix over the group ring *K*[*G*].
Let *G* = *G*_{1} > *G*_{2<
/sub>} > *G*_{3} > ··· be a chain
of normal subgroups of _G_ of finite index with tr
ivial intersection. The multiplication on the righ
t side by _A_ induces linear maps\n\nφ_{i}
: *K*[*G*/*G*_{i}]^*n*
^ → *K*[*G*/*G*_{i}]^*m^\n*

(*v*_{1}\,...\,*v*_{n}) → (*v*_{1}\,...\,*v*_{n<
/sub>})_A_.\n\nWe are interested in properties
of the sequence {dim_{K} ker φ_{i} / |*G*:*G*_{i}|}. In par
ticular\, we would like to answer the following qu
estions.\n\n# Is there the limit lim_{i
→ ∞} dim_{K} ker φ_{i}
/ |*G*:*G*_{i}|?\n# If the limi
t exists\, how does it depend on the chain {*G*~~i~~~~}?\n# What is the range of possible
values for lim~~_{i → ∞} dim_{K} ker φ_{i} / |*G*:*G*_{
i}| for a given group _G_?\n\nIt turns ou
t that the answers on these questions are known fo
r many groups _G_ if _K_ is a number field\, less
known if _K_ is an arbitrary field of characterist
ic 0 and almost unknown if _K_ is a field of posit
ive characteristic.\nIn my talk I will give severa
l motivations to consider these questions\, descri
be the known results and present recent advances i
n the case where _K_ has characteristic 0.
LOCATION:Poynting Physics\, Large Lecture Theatre
CONTACT:David Craven
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