On train-track splitting sequences. (English) Zbl 1275.57029

Train-tracks are fundamental objects introduced by Thurston in his study of surface diffeomorphisms, corresponding to embedded trivalent graphs on a surface \(S\) with additional structure see [R.C. Penner and J. L. Harer, Combinatorics of train tracks. Annals of Mathematics Studies. 125. Princeton, NJ: Princeton University Press. (1992; Zbl 0765.57001)]. Basic operations of splitting or sliding can be performed on a given train-track to obtain new train-tracks. A sliding and splitting sequence \(\{ \tau_i \}_{i=0}^N\) is a sequence of train-tracks where each \(\tau_{i+1}\) is obtained from \(\tau_i\) by sliding or splitting. On the other hand, given a subsurface \(X \subset S\), one can define the subsurface projection \(\pi_X (\tau)\) of a train-track \(\tau\) in the curve complex \(\mathcal{C} (X)\).
In this article the authors study the behavior of subsurface projections of train-track splitting sequences. To do so, new notions are introduced, namely the induced track which generalize the notion of subsurface projection of curves, the notion of efficient position which allows to pin down the location of an induced track and finally the wide curves which are combinatorial analogues of curves of definite modulus on a Riemann surface.
Using these notions, a detailed structure theorem for subsurface projection of splitting sequence is given. From this, the authors deduce that a subsurface projection \(\{\pi_X (\tau_i)\}\) of a sliding and splitting sequence \(\{ \tau_i \}\) is a \(Q\)-quasi-geodesic in the curve complex \(\mathcal{C} (X)\), where \(Q = Q(S)\) is a constant depending only on the surface \(S\).
This also allows the authors to generalize a result of U. Hamenstädt [Invent. Math. 175, No. 3, 545–609 (2009; Zbl 1197.57003)], proving that train-track sliding and splitting sequences give quasi-geodesics in the train-track graph \(\mathcal{T} (S)\).


57M60 Group actions on manifolds and cell complexes in low dimensions
20F65 Geometric group theory
Full Text: DOI arXiv Euclid


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