On finiteness of the number of boundary slopes of immersed surfaces in 3-manifolds (Q2781357)

Let \((F,\partial F)\) be a compact orientable surface with boundary, and \((M,\partial M)\) a compact orientable 3-manifold with boundary. An immersion \(f:(F,\partial F)\to (M,\partial M)\) is said to be proper if it takes boundaries to boundaries, so that \(f(F)\cap \partial M=f (\partial F)\). A closed curve in \(F\) is essential if it is not homotopic to a point, and a proper arc is essential if it is not homotopic (rel boundary) into \(\partial F\). A proper immersion \(f:(F,\partial F)\to (M,\partial M)\) is essential if no essential closed curve in \(F\) is homotopically trivial in \(M\), and no essential proper arc in \(F\) can be homotoped in \(M\) (rel boundary) into \(\partial M\). Surfaces which are incompressible and boundary incompressible are essential. A closed curve \(c\) is said to be primitive if it is not homotopic to \(b^n\), where \(b\) is a closed curve and \(n>1\). Let \(c\) be an essential primitive loop on \(\partial M\). If there is a proper immersion \(f\) of an essential surface \(F\) into \(M\) such that each component of \(\partial F\) is homotopic to a multiple of \(c\), then \(c\) is called a boundary slope of \(F\). The authors prove that for any hyperbolic 3-manifold \(M\) with totally geodesic boundary, there are finitely many boundary slopes for essential immersed surfaces of a given genus. Moreover, there is a uniform bound for the number of such boundary slopes if the genus of \(\partial M\) is bounded from above.