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.