On the H-type deviation of step two Carnot groups
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| Format: | Preprint |
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2023
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| _version_ | 1866917737736962048 |
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| author | Nalon, Luca Tyson, Jeremy T. |
| author_facet | Nalon, Luca Tyson, Jeremy T. |
| contents | The H-type deviation, $δ({\mathbb G})$, of a step two Carnot group ${\mathbb G}$ quantifies the extent to which ${\mathbb G}$ deviates from the geometrically and algebraically tractable class of Heisenberg-type (H-type) groups. In an earlier paper, the author defined this notion and used it to provide new analytic characterizations for the class of H-type groups. In addition, a quantitative conjecture relating the H-type deviation to the behavior of the $\infty$-Laplacian of Folland's fundamental solution for the $2$-Laplacian was formulated; an affirmative answer to this conjecture would imply that all step two polarizable groups are of H-type. In this paper, we elucidate further properties of the H-type deviation. First, we show that $0\le δ({\mathbb G}) \le 1$ for all step two Carnot groups ${\mathbb G}$. Recalling that $δ({\mathbb F}_{2,m}) = \sqrt{(m-2)/m}$, where ${\mathbb F}_{2,m}$ is the free step two Carnot group of rank $m$, we conjecture that $δ({\mathbb G}) \le \sqrt{(m-2)/m}$ for all step two rank $m$ groups. We explicitly compute $δ({\mathbb G})$ when ${\mathbb G}$ is a product of Heisenberg groups and verify the conjectural upper bound for such groups, with equality if and only if ${\mathbb G}$ factors over the first Heisenberg group. We also prove the following rigidity statement: for each $m \ge 3$ there exists $δ_0(m)>0$ so that if ${\mathbb G}$ is a step two and rank $m$ Carnot group with $δ({\mathbb G}) < δ_0(m)$, then ${\mathbb G}$ enjoys certain algebraic properties characteristic of H-type groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_06076 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the H-type deviation of step two Carnot groups Nalon, Luca Tyson, Jeremy T. Differential Geometry 53C17 The H-type deviation, $δ({\mathbb G})$, of a step two Carnot group ${\mathbb G}$ quantifies the extent to which ${\mathbb G}$ deviates from the geometrically and algebraically tractable class of Heisenberg-type (H-type) groups. In an earlier paper, the author defined this notion and used it to provide new analytic characterizations for the class of H-type groups. In addition, a quantitative conjecture relating the H-type deviation to the behavior of the $\infty$-Laplacian of Folland's fundamental solution for the $2$-Laplacian was formulated; an affirmative answer to this conjecture would imply that all step two polarizable groups are of H-type. In this paper, we elucidate further properties of the H-type deviation. First, we show that $0\le δ({\mathbb G}) \le 1$ for all step two Carnot groups ${\mathbb G}$. Recalling that $δ({\mathbb F}_{2,m}) = \sqrt{(m-2)/m}$, where ${\mathbb F}_{2,m}$ is the free step two Carnot group of rank $m$, we conjecture that $δ({\mathbb G}) \le \sqrt{(m-2)/m}$ for all step two rank $m$ groups. We explicitly compute $δ({\mathbb G})$ when ${\mathbb G}$ is a product of Heisenberg groups and verify the conjectural upper bound for such groups, with equality if and only if ${\mathbb G}$ factors over the first Heisenberg group. We also prove the following rigidity statement: for each $m \ge 3$ there exists $δ_0(m)>0$ so that if ${\mathbb G}$ is a step two and rank $m$ Carnot group with $δ({\mathbb G}) < δ_0(m)$, then ${\mathbb G}$ enjoys certain algebraic properties characteristic of H-type groups. |
| title | On the H-type deviation of step two Carnot groups |
| topic | Differential Geometry 53C17 |
| url | https://arxiv.org/abs/2312.06076 |