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

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• Andrei Jaikin-Zapirain, Universidad Autónoma de Madrid
• Saturday 12 August 2017, 09:30-10:30
• Poynting Physics, Large Lecture Theatre.

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

φ_i: K[G/G_i]ⁿ → K[G/G_i]^m
(v₁,...,v_n) → (v₁,...,v_n)A.

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

1. Is there the limit lim_{i → ∞} dim_K ker φ_i / |G:G_i|?
2. If the limit exists, how does it depend on the chain {G_i}?
3. What is the range of possible values for lim_{i → ∞} dim_K ker φ_i / |G:G_i| 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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• Poynting Physics, Large Lecture Theatre

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