The first L²-Betti number and approximation in arbitrary characteristic

2012/06/03 by Mikhail A. Ershov, Mikhail Ershov, Ershov, Mikhail +2
Computer Science · Mathematics · #Advanced Graph Theory Research #Advanced Operator Algebra Research #FOS: Mathematics #Group Theory (math.GR) #Operator Algebras (math.OA) #Primary: 20F65 #Secondary: 46Lxx #math.GR #math.OA #msc:20F65 #msc:46Lxx #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1206.0474

15 pages, final version, to appear in Documenta

openalex publication_date 2012/06/03 · arxiv created 2014/02/24 · arxiv updated 2014/02/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 b₁(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 b₁^(2)(G), the first L²-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 b₁^(2)(G). In particular, we show that the limit of the sequence b₁(G_i;F_p)/[G:G_i] is greater than or equal to b₁^(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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