**SEMINARIO DE TEORÍA DE GRUPOS**

**Łukasz Grabowski **

**(Lancaster) **

**"The Atiyah problem for k-homology gradients" **

**Lugar: Aula 520, departamento de matemáticas de la UAM. **

**Dia y hora: Martes, 15 de noviembre, a las 11:30. **

**Abstract: Thanks to the Lueck approximation theorem, the L2-Betti numbers of a normal covering of CW-complexes can be defined as limits of dimensions of the ordinary rational homology of intermediate covers. This leads to an interesting generalization of L2-Betti numbers: instead of the rational homology, one can try to use homology with coefficients in an arbitrary field k. This results in so called k-homology gradients. They were first suggested by Farber and later studied extensively by Lackenby and other authors. I will talk about a recent joint work with Thomas Shick where we study variants of the Atiyah problem for k-homology gradients. While overall results are similar to the results about L2-Betti numbers, there are some surprising differences: it is impossible to obtain an irrational k-homology gradient when the fundamental group of the base space is the lamplighter group, and the field k is of positive characteristic. We also disprove a conjecture of A.Thom, by showing that in general the k-homology gradients do not stabilize as the characteristic of k tends to infinity. Finally, we point out a family of concrete group ring elements in positive characteristic for which we currently do not know whether the Lueck approximation holds. **