A combinatorial representation of ∂D2 using train tracks

Let M be a closed, orientable surface of genus ≥ 2 . For a train track τ on M , Takarajima (this volume) presents a property for a curve called quasi-transversality, which is the combinatorial analogue of geodesicity. On the universal covering D2 of M , a geodesic has limit points at the ideal boundary ∂D2 , so does an infinite quasi-transverse curve. We study them in relation with D2 and represent the ideal boundary as the set of equivalent classes of infinite quasi-transverse curves. omatic structure for the mapping class group of M is constructed by Mosher (1995), where only one calculating step is left inexplicit. The study here enables us to cover this step. In doing this, we classify infinite quasi-transverse triangles. All procedures are combinatorial.