The first $L^2$-Betti number and approximation in arbitrary characteristic
Documenta mathematica, Tome 19 (2014), pp. 313-331
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 [G:Gi]b1(Gi;K) of normalized first Betti numbers of Gi 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 Fp-approximation have a limit, when these limits coincide, when they are independent of the sequence (Gi) and how they are related to b1(2)(G). In particular, we prove the inequality limi→∞[G:Gi]b1(Gi;Fp)≥b1(2)(G) under the assumptions that ∩Gi=1 and each G/Gi is a finite p-group.
Classification :
20F65, 46Lxx
Mots-clés : first L\^2-Betti number, approximation in prime characteristic
Mots-clés : first L\^2-Betti number, approximation in prime characteristic
@article{10_4171_dm_448,
author = {Mikhail Ershov and Wolfgang L\"uck},
title = {The first $L^2${-Betti} number and approximation in arbitrary characteristic},
journal = {Documenta mathematica},
pages = {313--331},
year = {2014},
volume = {19},
doi = {10.4171/dm/448},
url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/448/}
}
Mikhail Ershov; Wolfgang Lück. The first $L^2$-Betti number and approximation in arbitrary characteristic. Documenta mathematica, Tome 19 (2014), pp. 313-331. doi: 10.4171/dm/448
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