PL minimal surfaces in 3-manifolds

A Geometric Proof of Stallings' Theorem on Groups with More than One End

“…This is achieved by a PL version of the Meeks-Yau rounding trick used by Jaco and Rubinstein [8]. The details of the method, used in a context similar to the one we need, can be found in Lemma 2.7 of [16].…”

“…In this section we will make use of a method initially developed by Jaco and Rubinstein [13] and Casson which strengthens the notion of a least weight track to that of a minimal track with the objective of ensuring that minimal tracks which intersect are coincident. It will follow that the G orbit of a minimal track is a G-equivariant pattern and the G action on the dual tree to this pattern will induce a splitting of G over the stabiliser of the minimal track.…”

Incompressible surfaces and the topology of 3-dimensional manifolds

“…Let / : L -• M be an immersed, incompressible surface, with g homotopic to / chosen to minimize the area of g(L), among all maps in the homotopy class of/. In [8] Note that all the results in [8] can be obtained by using the theory of PL minimal surfaces developed in [22].…”

Boundary curves of surfaces with the 4–plane property

“…We say that Π above has least weight if every disk in Π has least weight among all the disks in M with the same boundary. It follows from Theorem 5 of [18] or Theorem 3.4 of [10] that f can be chosen so that Π has least weight, and hence any translate of Π has least weight. By Theorem 8 of [18] (or Theorem 6.3 of [10]), if there is a map g in the homotopy class of f having the n-plane property, then we can choose f so that f is a normal surface with least weight, Π has least weight, and f also has the n-plane property.…”

“…We will suppose that the lift of f into M F is an embedding (note that this is automatic if f is least area in the smooth or PL sense [14,18]). Thus, the preimage of f (F ) in M consists of an embedded simply connected surface Π which covers F in M F and the translates of Π by π 1 (M ).…”