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In this article, we give a survey of results on L2-Betti numbers and their analogues in positive characteristic. The main emphasis is made on the Lück approximation conjecture and the strong Atiyah conjecture.
Let π be a finite p-group and
a finite field with q = pn elements. Denote by
the augmentation ideal of the group ring
[π]. We have found a surprising relation between the abelianization of 1 +
, the Bogomolov multiplier B0(π) of π and the number of conjugacy classes k(π) of π:
For each prime
we construct a family
-groups such that
tends to zero as
tends to infinity. This disproves a well-known conjecture that
for every nonabelian finite
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