The first $L^2$-Betti number and approximation in arbitrary characteristic

Abstract

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 $\mathbb{Q}$-approximation and ${\mathbb{F}}_p$-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 prove the inequality ${\lim }_{i\to \infty }\frac{b_1(G_i;{\mathbb{F}}_p)}{[G:G_i]}\ge b_1^{(2)}(G)$ under the assumptions that $\cap G_i=1$ and each $G/G_i$ is a finite $p$-group.