The First L-betti Number and Approximation in Arbitrary Characteristic
نویسنده
چکیده
Let G be a finitely generated group and G = G0 ⊇ G1 ⊇ G2 ⊇ · · · a descending chain of finite index normal subgroups of G. Given a field K, we consider the sequence b1(Gi;K) [G:Gi] of normalized first Betti numbers of Gi with coefficients in K, which we call a K-approximation for b (2) 1 (G), the first L2-Betti number of G. In this paper we address the questions of when Qapproximation and Fp-approximation have a limit, when these limits coincide, when they are independent of the sequence (Gi) and how they are related to b (2) 1 (G). In particular, we prove the inequality limi→∞ b1(Gi;Fp) [G:Gi] ≥ b (2) 1 (G) under the assumptions that ∩Gi = {1} and each G/Gi is a finite p-group.
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