Geodesic
The first $L^2$-Betti number and approximation in arbitrary characteristic
Documenta mathematica, Tome 19 (2014), pp. 313-331.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Let $G$ be a finitely generated group and $G = G_0 \supseteq G_1 \supseteq G_2 \supseteq \cdots$ a descending chain of finite index normal subgroups of $G$. Given a field $K$, we consider the sequence $\frac{b_1(G_i;K)}{[G:G_i]}$ 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 $\IQ$-approximation and $\IF_p$-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 prove the inequality $\lim_{i\to\infty} \frac{b_1(G_i;\IF_p)}{[G:G_i]}\geq b_1^{(2)}(G)$ under the assumptions that $\cap G_i={1}$ and each $G/G_i$ is a finite $p$-group.
Classification : 20F65, 46Lxx
Keywords: first L^2-Betti number, approximation in prime characteristic 
@article{DOCMA_2014__19__a35,
     author = {Ershov, Mikhail and L\"uck, Wolfgang},
     title = {The first $L^2${-Betti} number and approximation in arbitrary characteristic},
     journal = {Documenta mathematica},
     pages = {313--331},
     publisher = {mathdoc},
     volume = {19},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2014__19__a35/}
}
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Ershov, Mikhail; Lück, Wolfgang. The first $L^2$-Betti number and approximation in arbitrary characteristic. Documenta mathematica, Tome 19 (2014), pp. 313-331. http://geodesic.mathdoc.fr/item/DOCMA_2014__19__a35/