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Quantitative nonembeddability of groups of polynomial growth into uniformly convex spaces
Metric Geometry · Submitted 22 Jul 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Nonabelian simply connected nilpotent Lie groups and not virtually abelian finitely generated groups of polynomial growth do not quasi-isometrically embed into uniformly convex Banach spaces.
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Nonabelian simply connected nilpotent Lie groups and not virtually abelian finitely generated groups of polynomial growth do not quasi-isometrically embed into uniformly convex Banach spaces. We quantify this fact by showing that a ball of radius $r\ge 2$ in the aforementioned groups must incur bilipschitz distortion at least a constant multiple of $(\log r)^{1/q}$ into a $q(\ge 2)$-uniformly convex Banach space. This bound is sharp for the $L^p$ ($1<p<\infty$) spaces. We prove this by establishing ``vertical versus horizontal inequalities'' for functions from the aforementioned groups into uniformly convex spaces, using the vector-valued Littlewood--Paley--Stein theory approach of Lafforgue and Naor (2012). These inequalities are quantitative nonembeddability statements that any Lipschitz mapping from the aforementioned groups into a uniformly convex space quantitatively collapses along certain central subgroups. In the special case of mappings of Carnot groups into the $L^p$ ($1<p<\infty$) spaces, we prove that the quantitative collapse occurs on the commutator subgroup; this is in line with the qualitative Pansu--Semmes nonembeddability argument given by Cheeger and Kleiner (2006) and Lee and Naor (2006). We prove this by establishing a version of the classical Dorronsoro theorem on Carnot groups. Previously, in the setting of Heisenberg groups, F\"assler and Orponen (2019) established a one-sided Dorronsoro theorem with a restriction $0<\alpha<2$ on the range of exponents $\alpha$ of the Laplacian; this restriction does not appear in the commutative setting and is caused by their use of horizontal polynomials as approximants. We identify the correct class of approximant polynomials and prove the two-sided Dorronsoro theorem with the full range $0<\alpha<\infty$ of exponents in the general setting of Carnot groups, thus strengthening and extending the work of F\"assler and Orponen.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Nonabelian simply connected nilpotent Lie groups and not virtually abelian finitely generated groups of polynomial growth do not quasi-isometrically embed into uniformly convex Banach spaces. | result 0.73 | 0.54 | 0.88 | asserted 1.00 |
| s1 | We quantify this fact by showing that a ball of radius $r\ge 2$ in the aforementioned groups must incur bilipschitz distortion at least a constant multiple of $(\log r)^{1/q}$ into a $q(\ge 2)$-uniformly convex Banach space. | result 0.94 | 0.96 | 0.37 | asserted 1.00 |
| s2 | This bound is sharp for the $L^p$ ($1<p<\infty$) spaces. | result 0.91 | 0.81 | 0.12 | asserted 1.00 |
| s3 | We prove this by establishing ``vertical versus horizontal inequalities'' for functions from the aforementioned groups into uniformly convex spaces, using the vector-valued Littlewood--Paley--Stein theory approach of Lafforgue and Naor (2012). | method 0.99 | 0.91 | 0.06 | asserted 1.00 |
| s4 | These inequalities are quantitative nonembeddability statements that any Lipschitz mapping from the aforementioned groups into a uniformly convex space quantitatively collapses along certain central subgroups. | interpretation 0.80 | 0.85 | 0.15 | asserted 1.00 |
| s5 | In the special case of mappings of Carnot groups into the $L^p$ ($1<p<\infty$) spaces, we prove that the quantitative collapse occurs on the commutator subgroup; this is in line with the qualitative Pansu--Semmes nonembeddability argument given by Cheeger and Kleiner (2006) and Lee and Naor (2006). | result 0.77 | 0.90 | 0.07 | asserted 0.98 |
| s6 | We prove this by establishing a version of the classical Dorronsoro theorem on Carnot groups. | method 0.96 | 0.94 | 0.03 | asserted 1.00 |
| s7 | Previously, in the setting of Heisenberg groups, F\"assler and Orponen (2019) established a one-sided Dorronsoro theorem with a restriction $0<\alpha<2$ on the range of exponents $\alpha$ of the Laplacian; this restriction does not appear in the commutative setting and is caused by their use of horizontal polynomials as approximants. | background 0.82 | 0.08 | 0.02 | asserted 1.00 |
| s8 | We identify the correct class of approximant polynomials and prove the two-sided Dorronsoro theorem with the full range $0<\alpha<\infty$ of exponents in the general setting of Carnot groups, thus strengthening and extending the work of F\"assler and Orponen. | contribution 0.88 | 0.97 | 0.07 | asserted 1.00 |
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Original abstract
Nonabelian simply connected nilpotent Lie groups and not virtually abelian finitely generated groups of polynomial growth do not quasi-isometrically embed into uniformly convex Banach spaces. We quantify this fact by showing that a ball of radius $r\ge 2$ in the aforementioned groups must incur bilipschitz distortion at least a constant multiple of $(\log r)^{1/q}$ into a $q(\ge 2)$-uniformly convex Banach space. This bound is sharp for the $L^p$ ($1<p<\infty$) spaces. We prove this by establishing ``vertical versus horizontal inequalities'' for functions from the aforementioned groups into uniformly convex spaces, using the vector-valued Littlewood--Paley--Stein theory approach of Lafforgue and Naor (2012). These inequalities are quantitative nonembeddability statements that any Lipschitz mapping from the aforementioned groups into a uniformly convex space quantitatively collapses along certain central subgroups. In the special case of mappings of Carnot groups into the $L^p$ ($1<p<\infty$) spaces, we prove that the quantitative collapse occurs on the commutator subgroup; this is in line with the qualitative Pansu--Semmes nonembeddability argument given by Cheeger and Kleiner (2006) and Lee and Naor (2006). We prove this by establishing a version of the classical Dorronsoro theorem on Carnot groups. Previously, in the setting of Heisenberg groups, F\"assler and Orponen (2019) established a one-sided Dorronsoro theorem with a restriction $0<\alpha<2$ on the range of exponents $\alpha$ of the Laplacian; this restriction does not appear in the commutative setting and is caused by their use of horizontal polynomials as approximants. We identify the correct class of approximant polynomials and prove the two-sided Dorronsoro theorem with the full range $0<\alpha<\infty$ of exponents in the general setting of Carnot groups, thus strengthening and extending the work of F\"assler and Orponen.
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What it cites · 6 sentences without a confident label
Uses the method or tool of 2 papers
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The following is an imitation of the argument of [ref] .
Foliated corona decompositions · The case $p<q$ · 0.89
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For $0<α≤1$ we again use characterizations of the fractional Laplacian given by [ref] for $0<α<1$ and [ref] for $α=1$ (subsubsection [ref]).
Mean Value Inequalities and Characterizations of Sobolev spaces on graded groups · Proof of Dorronsoro's Theorem [ref] for Carnot groups · 0.95
Uses data or values from 1 paper
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The exponent range is $[4,∞)$ for $G=ℍ^3$ [ref] and is $[2,∞)$ for $G=ℍ^k$, $k≥5$ [ref] .
Foliated corona decompositions · $L^1$-distortion: vertical perimeter versus horizontal perimeter · 0.88
Cites as background 1 paper
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See also Naor and Young [ref] for the endpoint case $p=1$ for the Heisenberg groups.
Foliated corona decompositions · Introduction · 0.83
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