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Asymptotic Dimension of Minor-Closed Families and Assouad-Nagata Dimension of Surfaces [article]

Marthe Bonamy, Nicolas Bousquet, Louis Esperet, Carla Groenland, Chun-Hung Liu, François Pirot, Alex Scott
2021 arXiv   pre-print
Assouad-Nagata dimension at most 2.  ...  We prove that every proper minor-closed family of graphs has asymptotic dimension at most 2, which gives optimal answers to a question of Fujiwara and Papasoglu and (in a strong form) to a problem raised  ...  The authors would like to thank Arnaud de Mesmay, Alfredo Hubard and Emil Saucan for the discussions on the discretization of Riemannian surfaces, and Gaël Meigniez for suggesting the simple proof of Lemma  ... 
arXiv:2012.02435v3 fatcat:hnmkgk3lcfaslj3weu7oblnaza

Surfaces have (asymptotic) dimension 2 [article]

Marthe Bonamy, Nicolas Bousquet, Louis Esperet, Carla Groenland, François Pirot, Alex Scott
2020 arXiv   pre-print
Finally we prove that the class of bounded degree graphs from any fixed proper minor-closed class has asymptotic dimension at most 2.  ...  This implies that the class of all graphs embeddable on any fixed surface (and in particular the class of planar graphs) has asymptotic dimension 2, which gives a positive answer to a recent question of  ...  The authors would like to thank Arnaud de Mesmay, Alfredo Hubard and Emil Saucan for the discussions on the discretization of Riemannian surfaces, and Gaël Meigniez for suggesting the simple proof of Lemma  ... 
arXiv:2007.03582v4 fatcat:gr67pautzbalvk4mwxexpnzfri

Asymptotic dimension of planes and planar graphs [article]

Koji Fujiwara, Panos Papasoglu
2021 arXiv   pre-print
In particular, the asymptotic dimension of the plane and any planar graph is at most three.  ...  We show that the asymptotic dimension of a geodesic space that is homeomorphic to a subset in the plane is at most three.  ...  We are grateful to the referee for very carefully reading the manuscripts and making precise and insightful comments.  ... 
arXiv:2002.01630v2 fatcat:5csukjmg25bqnm77jbmtf7z4y4

Triangulations of uniform subquadratic growth are quasi-trees [article]

Itai Benjamini, Agelos Georgakopoulos
2022 arXiv   pre-print
We also prove that every planar triangulation of asymptotic dimension 1 is quasi-isometric to a tree.  ...  The result extends to Riemannian 2-manifolds of finite genus, and to large-scale-simply-connected graphs.  ...  Acknowledgement We thank Panos Papazoglou and Federico Vigolo for helpful discussions. We thank Guy Lachman, Martin Winter and Geva Yashfe for helping us improve Problems 6.1 and 6.2.  ... 
arXiv:2106.06443v2 fatcat:2vzew5ytgzg73octgcxmh2awee

Higher rank hyperbolicity [article]

Bruce Kleiner, Urs Lang
2019 arXiv   pre-print
We prove a number of closely analogous results for spaces of rank n > 2 in an asymptotic sense, under some weak assumptions reminiscent of nonpositive curvature.  ...  Solving an asymptotic Plateau problem and producing unique tangent cones at infinity for such cycles, we show in particular that every quasi-isometry between two proper CAT(0) spaces of asymptotic rank  ...  or equal to the Assouad dimension).  ... 
arXiv:1810.12994v2 fatcat:2docuwugrnhkplwdofczr5dysq