×

zbMATH — the first resource for mathematics

On the ideal triangulation graph of a punctured surface. (English. French summary) Zbl 1256.32015
The authors define the ideal triangulation graph \(T(S)\) of a hyperbolic surface \(S\) by letting vertices be (isotopy classes of) ideal triangulations and placing edges between triangulations related by an elementary move (defined to be the operation of replacing an edge of the triangulation by another edge). The mapping class group \(\text{Mod}(S)\) acts on \(T(S)\), and thus induces an homomorphism \(\text{Mod}(S) \rightarrow \text{Aut}(T(S))\). They show that except for two low-complexity examples (punctured torus, thrice-punctured sphere, eliminating these examples is equivalent to the condition \(\chi(S) \leq -2\)), this map is an isomorphism. This is analogous to the situation for the curve complex or the arc complex. However, this graph is not Gromov hyperbolic, in contrast to the curve complex.

MSC:
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30F10 Compact Riemann surfaces and uniformization
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Gromov, M., Essays in Group Theory, edited by S.M. Gersten, Hyperbolic groups, 75-263, (1987), Springer-Verlag · Zbl 0634.20015
[2] Harer, J. L., Stability of the homology of the mapping class groups of orientable surfaces, Annals of Math., 121, 215-249, (1985) · Zbl 0579.57005
[3] Hatcher, A., On triangulations of surfaces, Top. and its Appl., 41, 189-194, (1991) · Zbl 0727.57012
[4] Irmak, E.; Korkmaz, M., Automorphisms of the hatcher-Thurston complex, Isr. J. Math., 162, 183-196, (2007) · Zbl 1149.57032
[5] Irmak, E.; McCarthy, J. D., Injective simplicial maps of the arc complex, Turkish Journal of Mathematics, 33, 1-16, (2009) · Zbl 1206.57018
[6] Ivanov, N. V., 1346, Automorphisms of Teichmüller modular groups, 199-270, (1988), Springer-Verlag, Berlin and New York · Zbl 0657.57004
[7] Ivanov, N. V.; McCarthy, J. D., On injective homomorphisms between Teichmüller modular groups, I. Invent. Math., 135, 2, 425-486, (1999) · Zbl 0978.57014
[8] Korkmaz, M., Automorphisms of complexes of curves on punctured spheres and on punctured tori, Topology and its Applications, 95, 2, 85-111, (1999) · Zbl 0926.57012
[9] Korkmaz, M.; Papadopoulos, A., On the arc and curve complex of a surface · Zbl 1194.57026
[10] Luo, F., Automorphisms of the complex of curves, Topology, 39, 2, 283-298, (2000) · Zbl 0951.32012
[11] Margalit, D., Automorphisms of the pants complex, Duke Math. J., 121, 3, 457-479, (2004) · Zbl 1055.57024
[12] Penner, R. C., The decorated Teichmüller space of punctured surfaces, Communications in Mathematical Physics, 113, 299-339, (1987) · Zbl 0642.32012
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.