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| Format: | Preprint |
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2025
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| Accès en ligne: | https://arxiv.org/abs/2512.17487 |
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| _version_ | 1866912776629256192 |
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| author | Bhowmik, Swarup Das, Deblina |
| author_facet | Bhowmik, Swarup Das, Deblina |
| contents | In this article, we study the algebraic and dynamical structure of certain normal subgroups of the quasi-isometry group of Euclidean space $QI(\mathbb{R}^n)$. For \[ H = \Big\{ [f] \in QI(\mathbb{R}^n) : \lim_{\|x\|\to\infty} \frac{\|f(x)-x\|}{\|x\|} = 0 \Big\}, \] and for $0<α<1$, \[ H_α= \Big\{ [f] \in QI(\mathbb{R}^n) : \|f(x)-x\| \le K\|x\|^α\text{ for large } \|x\| \Big\}, \] we show that each $H_α$ is a nontrivial normal subgroup of $QI(\mathbb{R}^n)$, satisfying \[ H_α\subset H_β\subset H \qquad \text{for } 0<α<β<1. \] We prove that the centers of $QI(\mathbb{R}^n)/H$ and $QI(\mathbb{R}^n)/H_α$ are trivial, while these quotients admit nontrivial torsion elements. Consequently, they are neither left-orderable nor locally indicable. Finally, we introduce an asymptotic topology on $QI(\mathbb{R}^n)$ and show that the family $\{H_α\}_{0<α<1}$ is dense in $H$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_17487 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Centers and Orderability of Certain Quotients of quasi-isometry groups of Euclidean spaces Bhowmik, Swarup Das, Deblina Geometric Topology 20F60, 20F65, 20F69 In this article, we study the algebraic and dynamical structure of certain normal subgroups of the quasi-isometry group of Euclidean space $QI(\mathbb{R}^n)$. For \[ H = \Big\{ [f] \in QI(\mathbb{R}^n) : \lim_{\|x\|\to\infty} \frac{\|f(x)-x\|}{\|x\|} = 0 \Big\}, \] and for $0<α<1$, \[ H_α= \Big\{ [f] \in QI(\mathbb{R}^n) : \|f(x)-x\| \le K\|x\|^α\text{ for large } \|x\| \Big\}, \] we show that each $H_α$ is a nontrivial normal subgroup of $QI(\mathbb{R}^n)$, satisfying \[ H_α\subset H_β\subset H \qquad \text{for } 0<α<β<1. \] We prove that the centers of $QI(\mathbb{R}^n)/H$ and $QI(\mathbb{R}^n)/H_α$ are trivial, while these quotients admit nontrivial torsion elements. Consequently, they are neither left-orderable nor locally indicable. Finally, we introduce an asymptotic topology on $QI(\mathbb{R}^n)$ and show that the family $\{H_α\}_{0<α<1}$ is dense in $H$. |
| title | Centers and Orderability of Certain Quotients of quasi-isometry groups of Euclidean spaces |
| topic | Geometric Topology 20F60, 20F65, 20F69 |
| url | https://arxiv.org/abs/2512.17487 |