“ Approximating L 2 - invariants by their finite - dimensional analogues ”

Let X be a finite connected CW -complex. Suppose that its fundamental group π is residually finite, i.e., there is a nested sequence . . . ⊂ Γm+1 ⊂ Γm ⊂ . . . ⊂ π of in π normal subgroups of finite index whose intersection is trivial. Then we show that the p-th L2-Betti number of X is the limit of the sequence bp(Xm)/[π : Γm] where bp(Xm) is the (ordinary) p-th Betti number of the finite covering of X associated with Γm. Introduction and statement of results Let X be a finite connected CW -complex with fundamental group π and A ⊂ X be a CW -subcomplex. Let b (2) p (X,A) be the p-th L-Betti number. Suppose π is residually finite, i.e., there is a nested sequence of in π normal subgroups . . . ⊂ Γm+1 ⊂ Γm ⊂ . . . ⊂ π such that the index [π : Γm] is finite for all m ≥ 0 and the intersection ∩m≥0Γm is the trivial group. Examples of residually finite groups are fundamental groups of compact Haken 3-manifolds and finitely generated groups possessing a faithful representation into GL(n, F ) for some field F . Consider any such sequence (Γm)m≥0. Let pm : Xm −→ X be the covering of X associated with Γm ⊂ π and put Am = p−1 m (A). Denote by bp(Xm, Am) the (ordinary) p-th Betti number of (Xm, Am). The following theorem answers a query of Gromov [14, pages 13, 153]. Theorem 0.1 (Kazhdan’s equality) Under the conditions above lim m→∞ bp(Xm, Am) [π : Γm] = b p (X,A). The inequality lim supm→∞ bp(Xm,Am) [π:Γm] ≤ b p (X,A) for X a closed manifold and empty A is discussed by Gromov [14, pages 13, 153] and is essentially due to Kazhdan [20]. There are estimates on the sum of the ordinary Betti numbers by the volume and by the simplicial volume for a closed Riemannian manifold which satisfies certain pinching