University of Birmingham > Talks@bham > Groups St Andrews 2017 > On ℓ^2-Betti numbers and their analogues in positive characteristic

On ℓ^2-Betti numbers and their analogues in positive characteristic

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If you have a question about this talk, please contact David Craven.

Let G be a group, K a field and A a n by m matrix over the group ring K[G]. Let G = G1 > G2 > G3 > ··· be a chain of normal subgroups of G of finite index with trivial intersection. The multiplication on the right side by A induces linear maps

φi: K[G/Gi]nK[G/Gi]m
(v1,...,vn) → (v1,...,vn)A.

We are interested in properties of the sequence {dimK ker φi / |G:Gi|}. In particular, we would like to answer the following questions.

  1. Is there the limit limi → ∞ dimK ker φi / |G:Gi|?
  2. If the limit exists, how does it depend on the chain {Gi}?
  3. What is the range of possible values for limi → ∞ dimK ker φi / |G:Gi| for a given group G?

It 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 if K is a field of positive characteristic. In my talk I will give several motivations to consider these questions, describe the known results and present recent advances in the case where K has characteristic 0.

This talk is part of the Groups St Andrews 2017 series.

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