# "On l2-Betti numbers and their analogues in positive characteristic" (Andrei Jaikin-Zapirain, Universidad Autonoma de Madrid)

25.04.2017 14:15

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 decreasing chain of normal subgroups of finite index of G, with trivial intersection. The multiplication on the right side by A induces linear maps from K[G/Gi]*n to K[G/Gi]*m, and we are interested in properties of the sequence of the dimensions of the kernels, with appropriate normalization. In particular, we would like to answer the following questions:

1) when does the limit exist ?

2) when the limit exists, how does it depend on the chain of the Gi ’s ?

3) for a given group G, what is the range of possible values of these limits ?

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.

### Lieu

Room 17, att. unusual place and time, Séminaire "Groupes et Géométrie"

### Organisé par

Section de mathématiques### Intervenants

Andrei Jaikin-Zapirain, Universidad Autonoma de Madridentrée libre

### Classement

Catégorie: Séminaire