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On the Lueck approximation conjectures

On the Lueck approximation conjectures



1.067 (20.30)


Andrei Jaikin




On the Lueck approximation conjectures

Abstract: The Lück approximation conjecture is a a collective term for a number of statements about approximations of L2 -Betti numbers.

A topological version of the conjecture can be formulated as follows. A group Γ satisfies the approximation conjecture if for every nested sequence Γ > Γ1 > Γ2 ... of normal subgroups of Γ with trivial intersection and for every G-CW-complex X of finite type
          bp(2) (X; l2(Γ)) = limi→∞ bp(2) (X/Γi; l2(Γ/Γi)) for every p ≥ 0.
The case where all the subgroups Γi are of finite index is a celebrated theorem of Wolfgang Lück published in 1994.

In my talk I will present an algebraic reformulation of this conjecture and several its generalizations which include the sofic Lück approximation conjecture and the Lück approximation conjecture in the space of marked groups. I will present the state of art of the subject and explain the solution of the Lück approximation conjecture when Γ is one-relator or locally indicable group.

Some of presented results are joint work with Diego López Álvarez.