Bounded generation of wreath products.
An abstract group \(G\) is said to have bounded generation if there exist (not necessarily distinct) elements \(g_1,\ldots,g_k\in G\) such that \(G=\langle g_1\rangle\cdots\langle g_k\rangle\). Even though defined as a simple combinatorial notion, bounded generation turns out to imply a number of remarkable structural properties: the pro-\(p\) completion of a boundedly generated group is a \(p\)-adic analytic group; if \(G\) is a boundedly generated \(S\)-arithmetic subgroup of an absolutely simple simply connected algebraic group over a number field, then \(G\) has the congruence subgroup property. In this paper the authors establish the following criterion for bounded generation of wreath products: if \(A\) and \(B\) are nontrivial groups, then \(A\wr B\) has bounded generation if and only if \(A\) has bounded generation and \(B\) is finite.
scientific article; zbMATH DE number 6506108