Injective maps between flip graphs
[Applications injectives entre graphes de triangulations]
Aramayona, Javier ; Koberda, Thomas ; Parlier, Hugo
Annales de l'Institut Fourier, Tome 65 (2015), p. 2037-2055 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

Nous montrons que, sauf dans quelques cas exceptionnels, toute application injective entre graphes de triangulations d’une surface est induite par une inclusion. Cela généralise un résultat de Korkmaz et Papadopoulos qui dit que tout automorphisme du graphe de triangulations d’une surface sans bord est induit par un homéomorphisme de la surface.

We prove that every injective simplicial map ℱ(S)→ℱ(S ' ) between flip graphs is induced by a subsurface inclusion S→S ' , except in finitely many cases. This extends a result of Korkmaz–Papadopoulos which asserts that every automorphism of the flip graph of a surface without boundary is induced by a surface homeomorphism.

Publié le : 2015-01-01
DOI : https://doi.org/10.5802/aif.2981
Classification:  57M50,  05C10,  05C60
Mots clés: Graphe des triangulations, flip, plongement 
@article{AIF_2015__65_5_2037_0,
     author = {Aramayona, Javier and Koberda, Thomas and Parlier, Hugo},
     title = {Injective maps between flip graphs},
     journal = {Annales de l'Institut Fourier},
     volume = {65},
     year = {2015},
     pages = {2037-2055},
     doi = {10.5802/aif.2981},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2015__65_5_2037_0}
}

Aramayona, Javier; Koberda, Thomas; Parlier, Hugo. Injective maps between flip graphs. Annales de l'Institut Fourier, Tome 65 (2015) pp. 2037-2055. doi : 10.5802/aif.2981. http://gdmltest.u-ga.fr/item/AIF_2015__65_5_2037_0/

[1] Aramayona, Javier Simplicial embeddings between pants graphs, Geom. Dedicata, Tome 144 (2010), pp. 115-128 | Article | Zbl 1194.57020

[2] Bell, M. An algorithm for deciding reducibility (http://arxiv.org/abs/1403.2997)

[3] Bridson, Martin R.; Haefliger, André Metric spaces of non-positive curvature, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 319 (1999), pp. xxii+643 | Article | Zbl 0988.53001

[4] Costantino, Francesco; Martelli, Bruno An analytic family of representations for the mapping class group of punctured surfaces, Geom. Topol., Tome 18 (2014) no. 3, pp. 1485-1538 | Article | Zbl 1311.57041

[5] Disarlo, V. Combinatorial rigidity of arc complexes (preprint)

[6] Disarlo, V.; Parlier, H. The geometry of flip graphs and mapping class groups (preprint)

[7] Irmak, Elmas; Mccarthy, John D. Injective simplicial maps of the arc complex, Turkish J. Math., Tome 34 (2010) no. 3, pp. 339-354 | Zbl 1206.57018

[8] Korkmaz, Mustafa; Papadopoulos, Athanase On the ideal triangulation graph of a punctured surface, Ann. Inst. Fourier (Grenoble), Tome 62 (2012) no. 4, pp. 1367-1382 http://aif.cedram.org/item?id=AIF_2012__62_4_1367_0 | Numdam | Zbl 1256.32015

[9] Masur, H. A.; Minsky, Y. N. Geometry of the complex of curves. II. Hierarchical structure, Geom. Funct. Anal., Tome 10 (2000) no. 4, pp. 902-974 | Article | Zbl 0972.32011