Injective maps between flip graphs
Annales de l'Institut Fourier, Volume 65 (2015) no. 5, pp. 2037-2055.

We prove that every injective simplicial map (S)(S ' ) between flip graphs is induced by a subsurface inclusion SS ' , 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.

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.

DOI: 10.5802/aif.2981
Classification: 57M50, 05C10, 05C60
Keywords: Triangulation graph, flip, embedding
Mot clés : Graphe des triangulations, flip, plongement
Aramayona, Javier 1; Koberda, Thomas 2; Parlier, Hugo 3

1 Université Paul Sabatier Institut de Mathématiques 118 Route de Narbonne 31000 Toulouse CEDEX 9 (France)
2 Yale University Department of Mathematics PO Box 208283 New Haven, CT 06520-8283 (USA)
3 University of Fribourg Department of Mathematics CH-1700 Fribourg (Switzerland)
@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},
     pages = {2037--2055},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {65},
     number = {5},
     year = {2015},
     doi = {10.5802/aif.2981},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2981/}
}
TY  - JOUR
AU  - Aramayona, Javier
AU  - Koberda, Thomas
AU  - Parlier, Hugo
TI  - Injective maps between flip graphs
JO  - Annales de l'Institut Fourier
PY  - 2015
SP  - 2037
EP  - 2055
VL  - 65
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2981/
DO  - 10.5802/aif.2981
LA  - en
ID  - AIF_2015__65_5_2037_0
ER  - 
%0 Journal Article
%A Aramayona, Javier
%A Koberda, Thomas
%A Parlier, Hugo
%T Injective maps between flip graphs
%J Annales de l'Institut Fourier
%D 2015
%P 2037-2055
%V 65
%N 5
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2981/
%R 10.5802/aif.2981
%G en
%F AIF_2015__65_5_2037_0
Aramayona, Javier; Koberda, Thomas; Parlier, Hugo. Injective maps between flip graphs. Annales de l'Institut Fourier, Volume 65 (2015) no. 5, pp. 2037-2055. doi : 10.5802/aif.2981. http://archive.numdam.org/articles/10.5802/aif.2981/

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

[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, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319, Springer-Verlag, Berlin, 1999, pp. xxii+643 | DOI | Zbl

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

[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., Volume 34 (2010) no. 3, pp. 339-354 | Zbl

[8] Korkmaz, Mustafa; Papadopoulos, Athanase On the ideal triangulation graph of a punctured surface, Ann. Inst. Fourier (Grenoble), Volume 62 (2012) no. 4, pp. 1367-1382 | EuDML | Numdam | Zbl

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

Cited by Sources: