Abstract
It is well-known that \(QI(\mathbb {R})\cong (QI(\mathbb {R}_{+})\times QI(\mathbb {R}_{-}))\rtimes <t>\), where \(QI(\mathbb {R})\)(resp. \(QI(\mathbb {R}_{+})(\cong QI(\mathbb {R_-}))\)) is the group of quasi-isometries of the real line (resp. \([0,\infty )\)). We introduce an invariant for the elements of \(QI(\mathbb {R_{+}})\) and split it into smaller units. We give an almost complete characterization of the elements of these units. We also show that a quotient of \(QI(\mathbb {R_{+}})\) gives an example of a left-orderable group which is not locally indicable.
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The authors thank Parameswaran Sankaran for his valuable suggestions and comments.
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The Swarup Bhowmik of this article acknowledges the financial support from Inspire, DST, Govt. of India as a Ph.D. student (Inspire) sanction letter number: No. DST/INSPIRE Fellowship/2018/IF180972.
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Bhowmik, S., Chakraborty, P. A structure theorem and left-orderability of a quotient of quasi-isometry group of the real line. Geom Dedicata 218, 12 (2024). https://doi.org/10.1007/s10711-023-00857-0
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DOI: https://doi.org/10.1007/s10711-023-00857-0