A combinatorial representation of \(\partial\mathbb{D}^2\) using train tracks (Q1579824)

The author continues his study of quasi-transversal curves with respect to a train track on a closed orientable surface \(M\) of genus \(g\geq 2\) in [ibid., 169-198 (2000; Zbl 0959.57016), see above]. He considers infinite quasi-transversal curves on the universal cover \(D^2\) of \(M\), he shows that such a curve has a unique limit point on \(\partial D^2\), and he gives a combinatorial algorithm to decide whether two curves have the same limit point. He then studies triangles bounded by quasi-transversal curves, in particular those with vertices in \(\partial D^2\), and obtains as an application the result that an automatic structure for the mapping class group of \(M\) can be explicitly constructed.