Large normal subgroup growth and large characteristic subgroup growth

Abstract

The maximal normal subgroup growth type of a finitely generated group is 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 ∆ a subgroup of finite index. Suppose ∆ has normal subgroup growth of type n log n , does Γ has normal subgroup growth of type 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 of 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 .2010 Mathematics subject classification. 20E07, 20E18, 20F05.