Large normal subgroup growth and large characteristic subgroup growth

Yiftach Barnea, J-C Schlage-Puchta

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The maximal normal subgroup growth type of a finitely generated group is nlogn. 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 nlogn, does Γ has normal subgroup growth of type nlogn? We give a positive answer in some cases, generalizing a result of M\"uller 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 nlogn. We also use our methods to show that one can find a group with characteristic subgroup growth of type n^logn.
Original languageEnglish
Pages (from-to)1-15
Number of pages15
JournalJournal of Group Theory
Issue number1
Early online date7 Sept 2019
Publication statusE-pub ahead of print - 7 Sept 2019

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