<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$s_n^\textrm{ch}(\Gamma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>s</mi> <mi>n</mi> <mtext>ch</mtext> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the number of characteristic subgroups of index at most <i>n</i> in a finitely generated group <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>. In response to a question of I. Rivin, we show that if <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Gamma = F_r$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Γ</mi> <mo>=</mo> <msub> <mi>F</mi> <mi>r</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> is the free group on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$r \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> generators, then the growth type of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$s_n^{\textrm{ch}}(F_r)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>s</mi> <mi>n</mi> <mtext>ch</mtext> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mi>r</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$n^{\textrm{log}(n)}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>n</mi> <mrow> <mtext>log</mtext> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> </math></EquationSource> </InlineEquation>. This is in contrast with the expectation of W. Thurston who predicted that there should be a difference between <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$r = 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$r > 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>></mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. 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.</p>

Characteristic subgroup growth

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2026年5月19日

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Liam Hanany,
Alexander Lubotzky

摘要

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 $r \ge 2$ $r \geq 2$ 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.