Characteristic subgroup growth

Abstract Let $$s_n^\textrm{ch}(\Gamma )$$ s n ch ( Γ ) denote the number of characteristic subgroups of index at most n in a finitely generated group $$\Gamma $$ Γ . In response to a question of I. Rivin, we show that if $$\Gamma = F_r$$ Γ = F r is the free group on $$r \ge 2$$ r ≥ 2 generators, then the growth type of $$s_n^{\textrm{ch}}(F_r)$$ s n ch ( F r ) is $$n^{\textrm{log}(n)}$$ n log ( n ) . This is in contrast with the expectation of W. Thurston who predicted that there should be a difference between $$r = 2$$ r = 2 and $$r > 2$$ r > 2 . Along the way, we answer a question of Barnea and Schlage-Puchta (J Group Theory 23(1):1–15, 2020) on the normal subgroup growth of large groups.