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

Let G be a group, K a field and A an nxm 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/Gi]n → K[G/Gi]m with Φ_i(v1,...,vn) = (v1,...,vn)A

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

1. Is there the limit $\lim_{i\to \infty}\frac{\dim_K \ker \phi_i}{|G:G_i|}$?

2. If the limit exists, how does it depend on the chain {G_i}?

3. What is the range of possible values for $\lim_{i\to \infty}\frac{\dim_K \ker \phi_i}{|G:G_i|}$ 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 Geometric Group Theory (GGT) Seminar series.