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Thompson’s group \(\mathcal T\) is the orientation-preserving automorphism group of a cellular complex. (English) Zbl 1292.57014

Summary: We consider a planar surface \(\Sigma\) of infinite type which has Thompson’s group \(\mathcal{T}\) as asymptotic mapping class group. We construct the asymptotic pants complex \(\mathcal{C}\) of \(\Sigma\) and prove that the group \(\mathcal{T}\) acts transitively by automorphisms on it. Finally, we establish that the automorphism group of the complex \(\mathcal{C}\) is an extension of the Thompson group \(\mathcal{T}\) by \(\mathbb{Z}/2\mathbb{Z}\)

MSC:

57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
20F38 Other groups related to topology or analysis
57M07 Topological methods in group theory
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
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[1] P. Bose and F. Hurtado, Flips in planar graphs, Comput. Geom. 42(1) (2009), 60\Ndash80. \smallDOI: 10.1016/j.comgeo.2008.04.001. · Zbl 1146.05016 · doi:10.1016/j.comgeo.2008.04.001
[2] M. G. Brin, The chameleon groups of Richard J. Thompson: automorphisms and dynamics, Inst. Hautes Études Sci. Publ. Math. 84(1) (1996), 5\Ndash33 (1997). \smallDOI: 10.1007/BF02698834. · Zbl 0891.57037 · doi:10.1007/BF02698834
[3] J. W. Cannon, W. J. Floyd, and W. R. Parry, Introductory notes on Richard Thompson’s groups, Enseign. Math. (2) 42(3-4) (1996), 215\Ndash256. \smallDOI: 10.5169/seals-87877. · Zbl 0880.20027
[4] S. L. Devadoss and R. C. Read, Cellular structures determined by polygons and trees, Ann. Comb. 5(1) (2001), 71\Ndash98. \smallDOI: 10.1007/PL00001293. · Zbl 0983.05008 · doi:10.1007/PL00001293
[5] D. S. Farley, Actions of picture groups on CAT(0) cubical complexes, Geom. Dedicata 110 (2005), 221\Ndash242. \smallDOI: 10.1007/s10711-004-1530-z. · Zbl 1139.20038 · doi:10.1007/s10711-004-1530-z
[6] L. Funar and C. Kapoudjian, On a universal mapping class group of genus zero, Geom. Funct. Anal. 14(5) (2004), 965\Ndash1012. \smallDOI: 10.1007/s00039-004-0480-9. · Zbl 1078.57021 · doi:10.1007/s00039-004-0480-9
[7] L. Funar and C. Kapoudjian, The braided Ptolemy-Thompson group is finitely presented, Geom. Topol. 12(1) (2008), 475\Ndash530. \smallDOI: 10.2140/gt.2008.12.475. · Zbl 1187.20029 · doi:10.2140/gt.2008.12.475
[8] L Funar and C. Kapoudjian, The braided Ptolemy-Thompson group is asynchronously combable, Comment. Math. Helv. 86(3) (2011), 707\Ndash768. \smallDOI: 10.4171/CMH/239. · Zbl 1266.57002 · doi:10.4171/CMH/239
[9] F. Haglund and F. Paulin, Simplicité de groupes d’automorphismes d’espaces à courbure négative, in: “The Epstein birthday schrift” , Geom. Topol. Monogr. 1 , Geom. Topol. Publ., Coven-try, 1998, pp. 181\Ndash248 (electronic). \smallDOI: 10.2140/gtm.1998.1.181. · Zbl 0916.51019 · doi:10.2140/gtm.1998.1.181
[10] F. Hurtado and M. Noy, Graph of triangulations of a convex polygon and tree of triangulations, Comput. Geom. 13(3) (1999), 179\Ndash188. \smallDOI: 10.1016/S0925-7721(99)00016-4. · Zbl 0948.68127 · doi:10.1016/S0925-7721(99)00016-4
[11] E. Irmak, Complexes of nonseparating curves and mapping classgroups, Michigan Math. J. 54(1) (2006), 81\Ndash110. \smallDOI: 10.1307/mmj/1144437439. · Zbl 1131.57019 · doi:10.1307/mmj/1144437439
[12] E. Irmak and M. Korkmaz, Automorphisms of the Hatcher-Thurston complex, Israel J. Math. 162 (2007), 183\Ndash196. \smallDOI: 10.1007/s11856-007-0094-7. · Zbl 1149.57032 · doi:10.1007/s11856-007-0094-7
[13] E. Irmak and J. D. McCarthy, Injective simplicial maps of the arc complex, Turkish J. Math. 34(3) (2010), 339\Ndash354. · Zbl 1206.57018
[14] N. V. Ivanov, Automorphism of complexes of curves and of Teichmüller spaces, Internat. Math. Res. Notices 14 (1997), 651\Ndash666. \smallDOI: 10.1155/S1073792897000433. · Zbl 0890.57018 · doi:10.1155/S1073792897000433
[15] C. Kapoudjian and V. Sergiescu, An extension of the Burau representation to a mapping class group associated to Thompson’s group \(T\), in: “Geometry and dynamics” , Contemp. Math. 389 , Amer. Math. Soc., Providence, RI, 2005, pp. 141\Ndash164. · Zbl 1138.20040 · doi:10.1090/conm/389/07277
[16] M. Korkmaz, Automorphisms of complexes of curves on punctured spheres and on punctured tori, Topology Appl. 95(2) (1999), 85\Ndash111. \smallDOI: 10.1016/S0166-8641(97)00278-2. · Zbl 0926.57012 · doi:10.1016/S0166-8641(97)00278-2
[17] M. Korkmaz and A. Papadopoulos, On the arc and curve complex of a surface, Math. Proc. Cambridge Philos. Soc. 148(3) (2010), 473\Ndash483. \smallDOI: 10.1017/S0305004109990387. · Zbl 1194.57026 · doi:10.1017/S0305004109990387
[18] C. W. Lee, The associahedron and triangulations of the \(n\)-gon, European J. Combin. 10(6) (1989), 551\Ndash560. · Zbl 0682.52004 · doi:10.1016/S0195-6698(89)80072-1
[19] P. Lochak and L. Schneps, The universal Ptolemy-Teichmüller groupoid, in: “Geometric Galois actions” , 2, London Math. Soc. Lecture Note Ser. 243 , Cambridge Univ. Press, Cambridge, 1997, pp. 325\Ndash347. \smallDOI: 10.1017/CBO9780511666124.014. · Zbl 0935.20026
[20] F. Luo, Automorphisms of the complex of curves, Topology 39(2) (2000), 283\Ndash298. \smallDOI: 10.1016/S0040-9383(99)00008-7. · Zbl 0951.32012 · doi:10.1016/S0040-9383(99)00008-7
[21] D. Margalit, Automorphisms of the pants complex, Duke Math. J. 121(3) (2004), 457\Ndash479. \smallDOI: 10.1215/S0012-7094-04-12133-5. · Zbl 1055.57024 · doi:10.1215/S0012-7094-04-12133-5
[22] P. Schmutz Schaller, Mapping class groups of hyperbolic surfaces and automorphism groups of graphs, Compositio Math. 122(3) (2000), 243\Ndash260. \smallDOI: 10.1023/A:1002672721132. · Zbl 0981.57004 · doi:10.1023/A:1002672721132
[23] D. D. Sleator, R. E. Tarjan, and W. P. Thurston, Rotation distance, triangulations, and hyperbolic geometry, J. Amer. Math. Soc. 1(3) (1988), 647\Ndash681. \smallDOI: 10.2307/1990951. · Zbl 0653.51017 · doi:10.2307/1990951
[24] J. Tits, Sur le groupe des automorphismes d’un arbre, in: “Essays on topology and related topics (Mémoires dédiés à Georges de Rham)” , Springer, New York, 1970, pp. 188\Ndash211. · Zbl 0214.51301
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