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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.

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30F10 Compact Riemann surfaces and uniformization
Full Text: DOI
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