Large normal subgroup growth and large characteristic subgroup growth

Abstract

Abstract The fastest normal subgroup growth type of a finitely generated group is n log ⁡ n {n^{\log n}} . Very little is known about groups with this type of growth. In particular, the following is a long standing problem: Let Γ be a group and Δ a subgroup of finite index. Suppose Δ has normal subgroup growth of type n log ⁡ n {n^{\log n}} . Does Γ have normal subgroup growth of type n log ⁡ n {n^{\log n}} ? We give a positive answer in some cases, generalizing a result of Müller and the second author and a result of Gerdau. For instance, suppose G is a profinite group and H an open subgroup of G. We show that if H is a generalized Golod–Shafarevich group, then G has normal subgroup growth of type n log ⁡ n {n^{\log n}} . We also use our methods to show that one can find a group with characteristic subgroup growth of type n log ⁡ n {n^{\log n}} .