Abstract: The Lück approximation conjecture is a a collective term for a number of statements about approximations of L^{2} -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 > Γ_{1} > Γ_{2} ... of normal subgroups of Γ with trivial intersection and for every G-CW-complex X of finite type

b_{p}^{(2)} (X; l^{2}(Γ)) = lim_{i→∞} b_{p}^{(2)} (X/Γ_{i}; l^{2}(Γ/Γ_{i})) for every p ≥ 0.

The case where all the subgroups Γ_{i} 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.