Geodesic
Asymptotic properties of the Heisenberg group
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 4, Tome 261 (1999), pp. 125-154

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It is shown that Hausdorff–Gromov distance between the discrete Heisenberg group with word metric and its asymptotic cone is finite 
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     author = {S. A. Krat},
     title = {Asymptotic properties of the {Heisenberg} group},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_1999_261_a9/}
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S. A. Krat. Asymptotic properties of the Heisenberg group. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 4, Tome 261 (1999), pp. 125-154. http://geodesic.mathdoc.fr/item/ZNSL_1999_261_a9/