The Cohen-Lenstra heuristics [H. Cohen and H. W. Lenstra jun., Lect. Notes Math. 1068, 33–62 (1984; Zbl 0558.12002)] conjectured that a particular finite abelian group should occur as the class group of a quadratic imaginary field with frequency inversely proportional to its number of automorphisms. The paper under review was motivated by the Cohen-Lenstra heuristics over function fields. In particular, the authors prove that for \(\ell > 2\) a prime and \(A\) a finite abelian \(\ell\)-group, there exists \(Q = Q(A)\) such that for \(q\) greater than \(Q\), a positive fraction of quadratic extensions of \(\mathbb F_q(t)\) have the \(\ell\)-part of their class group isomorphic to \(A\). The key ingredient in the proof surprisingly uses stable homology of Hurwitz spaces. More precisely, let \(c\) be a conjugacy class in a finite group \(G\). The Hurwitz space \(\mathrm{Hur}^c_{G, n}\) parameterizes \(G\)-covers of the complex projective line with \(n\) branch points, each of which has monodromy belonging to the conjugacy class \(c\). Assume that \(c\) generates \(G\) and that for any subgroup \(H\leq G\), the intersection of \(c\) with \(H\) is either empty or a conjugacy class of \(H\). Then the authors show that there exist integers \(A, B, D > 0\) such that \(\mathrm{Hur}^c_{G,n}\) and \(\mathrm{Hur}^c_{G,n+D}\) have the same \(p\)th Betti number whenever \(n\geq Ap + B\). A number of interesting consequences, related work, and conjectures are also discussed in detail.