On the peripheral subgroups of irreducible 3-manifold groups and acylindrical splittings

Abstract

We survey the problem of separation under conjugacy and malnormality of the abelian peripheral subgroups of an orientable, irreducible $3$-manifold $X$. We shall focus on the relation between this problem and the existence of acylindrical splittings of $\pi_1(X)$ as an amalgamated product or HNN-extension along the abelian subgroups corresponding to the JSJ-tori.