The First L^2-Betti Number and Approximation in Arbitrary Characteristic

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.

2010 Mathematics Subject Classification: Primary: 20F65; Secondary: 46Lxx

Keywords and Phrases: First L^2-Betti number, approximation in prime characteristic

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