第一$L^2$ -贝蒂数及任意特征的逼近
The first $L^2$-Betti number and approximation in arbitrary characteristic
摘要：
设G是一个有限生成群，（G_i）是G的有限指数正规子群的下降链.给定一个域K，我们考虑G_i的系数在K中的正规化第一Betti数序列b_1（G_i;K）/[G：G_i]，我们称之为G的第一L^2-贝蒂数b_1^（2）（G）的K-逼近.本文讨论了Q-逼近和F_p-逼近何时有极限，何时这些极限重合，何时它们与序列（G_i）无关，以及它们与b_1^（2）（G）的关系。特别地，在（G_i）有平凡交且每个G/G_i是有限p-群的条件下，证明了序列b_1（G_i;F_p）/[G：G_i]的极限大于或等于b_1^（2）（G）.
英文摘要：
Let G be a finitely generated group and (G_i) a descending chain of finite index normal subgroups of G. Given a field K, we consider the sequence b_1(G_i;K)/[G:G_i] of normalized first Betti numbers of G_i with coefficients in K, which we call a K-approximation for b_1^(2)(G), the first L^2-Betti number of G. In this paper we address the questions of when Q-approximation and F_p-approximation have a limit, when these limits coincide, when they are independent of the sequence (G_i) and how they are related to b_1^(2)(G). In particular, we show that the limit of the sequence b_1(G_i;F_p)/[G:G_i] is greater than or equal to b_1^(2)(G) under the assumptions that (G_i) has trivial intersection and each G/G_i is a finite p-group.
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