Non-compact totally peripheral 3-manifolds (Q1891200)

A 3-manifold is totally peripheral if every loop is freely homotopic into the boundary. \textit{M. Brin, K. Johannson} and \textit{P. Scott} [Pac. J. Math. 118, 37-51 (1985; Zbl 0525.57010)] studied compact totally peripheral 3-manifolds. They showed that when \(M\) is orientable, compact and totally peripheral, then there is a boundary component \(F\) of \(M\) such that the natural map \(\pi_ 1(F) \to \pi_ 1(M)\) is surjective. When \(M\) is non-orientable, they showed that this result is almost true but that there are essentially two counterexamples. In this paper, we show that the same results hold if the compactness hypothesis on \(M\) is omitted. The results remain true even if the fundamental group of \(M\) is not finitely generated.