Characteristic Subgroup Growth

TL;DR

The paper resolves Rivin's question by showing that the characteristic subgroup growth of the free group F_r (r≥2) is of type $n^{\log n}$, identical to its normal subgroup growth and contrary to Thurston's expectation of divergence between ranks. The authors leverage the Zassenhaus filtration to obtain exponential growth in the dimensions of the associated graded pieces, then translate this into large numbers of characteristic subgroups via lifting of Aut(F_r)-invariant submodules. They further construct a group where characteristic and normal subgroup growth types differ, using Segal-style wreath products and a dense subgroup with congruence properties, thereby showing these growth types need not coincide in general. The work also discusses extensions to large groups and lattices, offering insights and open questions about when characteristic and normal growth align, with implications for rigidity phenomena in lattices and automorphism groups.

Abstract

Let $s_n^\mathrm{ch}(Γ)$ denote the number of characteristic subgroups of index at most $n$ in a finitely generated group $Γ$. In response to a question of I. Rivin we show that if $Γ= F_r$ is the free group on $r \geq 2$ generators then the growth type of $s_n^{\mathrm{ch}}(F_r)$ is $n^{\mathrm{log}(n)}$. This is in contrast with the expectation of W. Thurston who predicted that there should be a difference between $r = 2$ and $r > 2$. Along the way we answer a question of arXiv:1703.07866 on the normal subgroup growth of large groups.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Proposition 1.3
• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• Lemma 2.4
• Corollary 2.5
• Proposition 2.6