Let M be an open (complete and non-compact) manifold with <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\textrm{Ric}\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Ric</mtext> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and escape rate not 1/2. It is known that under these conditions, the fundamental group <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\pi _1(M)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>π</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> has a finitely generated torsion-free nilpotent subgroup <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">N</mi> </math></EquationSource> </InlineEquation> of finite index, as long as <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\pi _1(M)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>π</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is an infinite group. We show that the nilpotency step of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">N</mi> </math></EquationSource> </InlineEquation> must be reflected in the asymptotic geometry of the universal cover <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\widetilde{M}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>M</mi> <mo stretchy="false">~</mo> </mover> </math></EquationSource> </InlineEquation>, in terms of the Hausdorff dimension of an isometric <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation>-orbit: there exist an asymptotic cone (Y, y) of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\widetilde{M}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>M</mi> <mo stretchy="false">~</mo> </mover> </math></EquationSource> </InlineEquation> and a closed <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation>-subgroup L of the isometry group of Y such that its orbit Ly has Hausdorff dimension at least the nilpotency step of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">N</mi> </math></EquationSource> </InlineEquation>. This resolves a question raised by Wei and the author (see Pan and Wei in Geom Funct Anal 32:676–685, 2022, Remark 1.7 and Pan in Geom Topol 28:1409–1436, 2024, Conjecture 0.2).

Nonnegative Ricci curvature, nilpotency, and Hausdorff dimension

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2026年1月1日

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Jiayin Pan

摘要

Let M be an open (complete and non-compact) manifold with $\textrm{Ric}\ge 0$ $Ric \geq 0$ and escape rate not 1/2. It is known that under these conditions, the fundamental group $\pi _1(M)$ $π_{1} (M)$ has a finitely generated torsion-free nilpotent subgroup $\mathcal {N}$ $N$ of finite index, as long as $\pi _1(M)$ $π_{1} (M)$ is an infinite group. We show that the nilpotency step of $\mathcal {N}$ $N$ must be reflected in the asymptotic geometry of the universal cover $\widetilde{M}$ $\tilde{M}$ , in terms of the Hausdorff dimension of an isometric $\mathbb {R}$ $R$ -orbit: there exist an asymptotic cone (Y, y) of $\widetilde{M}$ $\tilde{M}$ and a closed $\mathbb {R}$ $R$ -subgroup L of the isometry group of Y such that its orbit Ly has Hausdorff dimension at least the nilpotency step of $\mathcal {N}$ $N$ . This resolves a question raised by Wei and the author (see Pan and Wei in Geom Funct Anal 32:676–685, 2022, Remark 1.7 and Pan in Geom Topol 28:1409–1436, 2024, Conjecture 0.2).