The first \(L^2\)-Betti number and approximation in arbitrary characteristic (Q2439236)

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 \(\mathbb Q\)-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 \(\lim_{i\to\infty} \frac{b_1(G_i;\mathbb F_p)}{[G:G_i]}\geq b_1^{(2)}(G)\) under the assumptions that \(\cap G_i={1}\) and each \(G/G_i\) is a finite \(p\)-group.