×

zbMATH — the first resource for mathematics

Simplicial embeddings between pants graphs. (English) Zbl 1194.57020
Let \(\Sigma\) be a compact orientable surface such that every connected component has negative Euler characteristic. When the genus of \(\Sigma\) is \(g\) and the number of boundary components of \(\Sigma\) is \(b\), we define the complexity of \(\Sigma\) by \(\kappa(\Sigma) = 3g-3+b\). A curve on \(\Sigma\) is a homotopy class of nontrivial and nonperipheral simple closed curves on \(\Sigma\). A multicurve \(Q\) on \(\Sigma\) is a collection of pairwise distinct and pairwise disjoint curves on \(\Sigma\). A pants decomposition \(Q\) of \(\Sigma\) is a multicurve \(Q\) such that \(|Q| = \kappa(\Sigma)\), i.e. each connected component of \(\Sigma - Q\) is a \(3\)-holed sphere. The pants graph \({\mathcal P}(\Sigma)\) of \(\Sigma\) is the simplicial graph whose vertex set is the set of all pants decompositions of \(\Sigma\) and where two vertices are connected by an edge if the corresponding pants decompositions are related by an elementary move.
The pants graph was introduced by A. E. Hatcher and W. Thurston [Topology 19, 221–237 (1980; Zbl 0447.57005)], who proved it is connected. D. Margalit [Duke Math. J. 121, No. 3, 457–479 (2004; Zbl 1055.57024)] showed that if \(\Sigma\) is a compact connected oriented surface with \(\kappa(\Sigma) > 0\), then the natural homomorphism from the mapping class group of \(\Sigma\) to the automorphism group of \({\mathcal P}(\Sigma)\) is injective, moreover, if \(\kappa(\Sigma)>3\), then this homomorphism is an isomorphism.
In the paper under review, as an extension of the above result by Margalit to injective simplicial maps between pants graphs, it is shown: let \(\Sigma_1\) and \(\Sigma_2\) be compact orientable surfaces of negative Euler characteristic such that each connected component of \(\Sigma_1\) has complexity at least \(2\), and let \(\phi : {\mathcal P}(\Sigma_1) \to {\mathcal P}(\Sigma_2)\) be an injective simplicial map, then there exists a \(\pi_1\)-injective embedding \(h : \Sigma_1 \to \Sigma_2\) and a multicurve \(Q\) on \(\Sigma_2\) such that \(\phi(v) = h(v) \cup Q\) for all vertices \(v\) of \({\mathcal P}(\Sigma_1)\). This result is proved by investigation on a stratified structure \(\{ {\mathcal P}_Q ; Q \text{ is a multicurve on } \Sigma \}\) of \({\mathcal P}(\Sigma)\), where \({\mathcal P}_Q\) is a subgraph of \({\mathcal P}(\Sigma)\) spanned by vertices of \({\mathcal P}(\Sigma)\) that contains \(Q\), especially on Farey graphs (\({\mathcal P}_Q\) for the multicurve \(Q\) of cardinality \(\kappa(\Sigma)-1\)) in \({\mathcal P}(\Sigma)\).

MSC:
57M50 General geometric structures on low-dimensional manifolds
05C60 Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Aramayona, J., Leininger, C.J., Souto, J.: Injections of mapping class groups of closed surfaces. Geom. Topol. (to appear) · Zbl 1225.57001
[2] Behrstock J., Margalit D.: Curve complexes and finite index subgroups of mapping class groups. Geom. Dedicata 118(1), 71–85 (2006) · Zbl 1129.57023 · doi:10.1007/s10711-005-9022-3
[3] Bell R., Margalit D.: Braid groups and the co-Hopfian property. J. Algebra 303, 275–294 (2006) · Zbl 1110.20028 · doi:10.1016/j.jalgebra.2005.10.038
[4] Birman J., Lubotzky A., McCarthy J.: Abelian and solvable subgroups of the mapping class groups. Duke Math. J. 50(4), 1107–1120 (1983) · Zbl 0551.57004 · doi:10.1215/S0012-7094-83-05046-9
[5] Brock J.F.: The Weil-Petersson metric and volumes of 3-dimensional hyperbolic convex cores. J. Am. Math. Soc. 16, 495–535 (2003) · Zbl 1059.30036 · doi:10.1090/S0894-0347-03-00424-7
[6] Brock J.F., Margalit D.: Weil-Petersson isometries via the pants graph. Proc. Am. Math. Soc. 135(3), 795–803 (2007) · Zbl 1110.32004 · doi:10.1090/S0002-9939-06-08577-7
[7] Harer J.: The virtual cohomological dimension of the mapping class group of an orientable surface. Invent. Math. 84(1), 157–176 (1986) · Zbl 0592.57009 · doi:10.1007/BF01388737
[8] Hatcher A., Lochak P., Schneps L.: On the Teichmüller tower of mapping class groups. J. reine angew. Math. 521, 1–24 (2000) · Zbl 0953.20030 · doi:10.1515/crll.2000.028
[9] Hatcher A.E., Thurston W.P.: A presentation for the mapping class group of a closed orientable surface. Topology 19, 221–237 (1980) · Zbl 0447.57005 · doi:10.1016/0040-9383(80)90009-9
[10] Irmak E.: Superinjective simplicial maps of complexes of curves and injective homomorphisms of subgroups of mapping class groups. Topology 43(3), 513–541 (2004) · Zbl 1052.57024 · doi:10.1016/j.top.2003.03.002
[11] Irmak E.: Superinjective simplicial maps of complexes of curves and injective homomorphisms of subgroups of mapping class groups II. Topol. Appl. 153, 1309–1340 (2006) · Zbl 1100.57020 · doi:10.1016/j.topol.2005.04.001
[12] Irmak E.: Complexes of nonseparating curves and mapping class groups. Mich. Math. J. 54, 81–110 (2006) · Zbl 1131.57019 · doi:10.1307/mmj/1144437439
[13] Irmak E., Korkmaz M.: Automorphisms of the Hatcher–Thurston complex. Isr. J. Math. 162, 183–196 (2007) · Zbl 1149.57032 · doi:10.1007/s11856-007-0094-7
[14] Irmak, E., McCarthy, J.D.: Injective simplicial maps of the arc complex”, (with McCarthy J.D.). Preprint, arXiv:math/0606612 · Zbl 1206.57018
[15] Ivanov N.V.: Automorphism of complexes of curves and of Teichmüller spaces. Intern. Math. Res. Not. 14, 651–666 (1997) · Zbl 0890.57018 · doi:10.1155/S1073792897000433
[16] Ivanov N.V., McCarthy J.D.: On injective homomorphisms between Teichmüller modular groups I. Invent. Math. 135, 425–486 (1999) · Zbl 0978.57014 · doi:10.1007/s002220050292
[17] Korkmaz M.: Automorphisms of complexes of curves on punctured spheres and on punctured tori. Topol. Appl. 95(2), 85–111 (1999) · Zbl 0926.57012 · doi:10.1016/S0166-8641(97)00278-2
[18] Luo F.: Automorphisms of the complex of curves. Topology 39(2), 283–298 (2000) · Zbl 0951.32012 · doi:10.1016/S0040-9383(99)00008-7
[19] Margalit D.: Automorphisms of the pants complex. Duke Math. J. 121(3), 457–479 (2004) · Zbl 1055.57024 · doi:10.1215/S0012-7094-04-12133-5
[20] Masur H., Wolf M.: The Weil-Petersson isometry group. Geom. Dedicata 93, 177–190 (2002) · Zbl 1014.32008 · doi:10.1023/A:1020300413472
[21] Minsky, Y.N.: A geometric approach to the complex of curves on a surface. In: Topology and Teichmüller spaces (Katinkulta, 1995), pp. 149–158. World Science Publishing, River Edge (1996) · Zbl 0937.30027
[22] Paris L., Rolfsen D.: Geometric subgroups of mapping class groups. J. Reine Angew. Math. 521, 47–83 (2000) · Zbl 1007.57014 · doi:10.1515/crll.2000.030
[23] Powell J.: Two theorems on the mapping class group of a surface. Proc. Am. Math. Soc. 68(3), 347–350 (1978) · Zbl 0391.57009 · doi:10.1090/S0002-9939-1978-0494115-8
[24] Shackleton, K.J.: Combinatorial rigidity in curve complexes and mapping class groups. Pac. J. Math. 230(1), 2007 · Zbl 1165.57017
[25] Schmutz-Schaller P.: Mapping class groups of hyperbolic surfaces and automorphism groups of graphs. Comp. Math. 122, 243–260 (2000) · Zbl 0981.57004 · doi:10.1023/A:1002672721132
[26] Wolpert, S.A.: Geometry of the Weil-Petersson completion of Teichmueller space. In: Yau, S.T. Surveys in Differential Geometry, VIII: Papers in Honor of Calabi, Lawson, Siu and Uhlenbeck, International Press, Cambridge (2003) · Zbl 1049.32020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.