flip graphs; infinite type surfaces; Triangulations; Geometry and Topology; Discrete Mathematics and Combinatorics
Abstract :
[en] We associate to triangulations of infinite type surfaces a type of flip graph where simultaneous flips are allowed. Our main focus is on understanding exactly when two triangulations can be related by a sequence of flips. A consequence of our results is that flip graphs for infinite type surfaces have uncountably many connected components.
Disciplines :
Mathematics
Author, co-author :
Fossas, Ariadna ✱; GAP Nonlinearity and Climate, Institut des Sciences de l’Environnement, Université de Genève, Genève, Switzerland
PARLIER, Hugo ✱; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)
✱ These authors have contributed equally to this work.
External co-authors :
yes
Language :
English
Title :
Flip graphs for infinite type surfaces
Publication date :
2022
Journal title :
Groups, Geometry, and Dynamics
ISSN :
1661-7207
eISSN :
1661-7215
Publisher :
European Mathematical Society Publishing House
Volume :
16
Issue :
4
Pages :
1165 - 1178
Peer reviewed :
Peer reviewed
Funding text :
Funding. The second author was supported by the Luxembourg National Research Fund OPEN grant O19/13865598.
J. Aramayona, A. Fossas, and H. Parlier, Arc and curve graphs for infinite-type surfaces. Proc. Amer. Math. Soc. 145 (2017), no. 11, 4995–5006 Zbl 1377.57006 MR 3692012
A. Basmajian, Hyperbolic structures for surfaces of infinite type. Trans. Amer. Math. Soc. 336 (1993), no. 1, 421–444 Zbl 0768.30028 MR 1087051
A. Basmajian, Large parameter spaces of quasiconformally distinct hyperbolic structures. J. Anal. Math. 71 (1997), 75–85 Zbl 0891.30025 MR 1454244
A. Basmajian and Y. Kim, Geometrically infinite surfaces with discrete length spectra. Geom. Dedicata 137 (2008), 219–240 Zbl 1167.30022 MR 2449153
J. Bavard, Hyperbolicité du graphe des rayons et quasi-morphismes sur un gros groupe modulaire. Geom. Topol. 20 (2016), no. 1, 491–535 Zbl 1362.37086 MR 3470720
P. Bose, J. Czyzowicz, Z. Gao, P. Morin, and D. R. Wood, Simultaneous diagonal flips in plane triangulations. J. Graph Theory 54 (2007), no. 4, 307–330 Zbl 1120.05024 MR 2292668
R. L. Brooks, On colouring the nodes of a network. Proc. Cambridge Philos. Soc. 37 (1941), 194–197 Zbl 0027.26403 MR 12236
V. Disarlo and H. Parlier, Simultaneous flips on triangulated surfaces. Michigan Math. J. 67 (2018), no. 3, 451–464 Zbl 1402.57018 MR 3835560
V. Disarlo and H. Parlier, The geometry of flip graphs and mapping class groups. Trans. Amer. Math. Soc. 372 (2019), no. 6, 3809–3844 Zbl 1423.57004 MR 4009420
M. G. Durham, F. Fanoni, and N. G. Vlamis, Graphs of curves on infinite-type surfaces with mapping class group actions. Ann. Inst. Fourier (Grenoble) 68 (2018), no. 6, 2581–2612 Zbl 1416.57013 MR 3897975
F. Fanoni, T. Ghaswala, and A. McLeay, Homeomorphic subsurfaces and the omnipresent arcs. Ann. H. Lebesgue 4 (2021), 1565–1593 Zbl 1490.57014 MR 4353971
A. Fossas and M. Nguyen, Thompson’s group T is the orientation-preserving automorphism group of a cellular complex. Publ. Mat. 56 (2012), no. 2, 305–326 Zbl 1292.57014 MR 2978326
L. Liu and A. Papadopoulos, Some metrics on Teichmüller spaces of surfaces of infinite type. Trans. Amer. Math. Soc. 363 (2011), no. 8, 4109–4134 Zbl 1260.32003 MR 2792982
A. McLeay and H. Parlier, Ideally, all infinite type surfaces can be triangulated. Bull. Lond. Math. Soc., to appear
L. Mosher, Mapping class groups are automatic. Ann. of Math. (2) 142 (1995), no. 2, 303–384 Zbl 0867.57004 MR 1343324
H. Parlier and A. Weber, Geometric simplicial embeddings of arc-type graphs. J. Korean Math. Soc. 57 (2020), no. 5, 1103–1118 Zbl 1482.57022 MR 4169559
R. C. Penner and D. Šarić, Teichmüller theory of the punctured solenoid. Geom. Dedicata 132 (2008), 179–212 Zbl 1182.30076 MR 2396916
