Brick's Quasi Simple Filtrations and 3-Manifolds
John R. Stallings · Cambridge University Press eBooks · 1993
. Poénaru [P] and Casson have developed an idea about the metric geometry of the Cayley graph of a group having to do with certain problems in 3-manifold theory; this is described in [G] and in [GS]. Brick [B] has developed a non-metric condition related to this, which he calls “quasi simple filtration”; a space is qsf if it is, approximately, the union of an increasing sequence of compact, 1-connected spaces. Here we outline these notions and establish the theory in a polyhedral setting. This provides a purely group-theoretic notion of qsf which seems interesting in itself. Polyhedral niceties B asic facts : The theory of finite polyhedra is a standard subject; one reference is [AH], Chapter 3. The constructions in Whitehead's paper [W] involve simplicial complexes and subdivisions of particular sorts, and this too can be considered a polyhedral reference. We shall outline some of the theory here. A finite polyhedron P is a subset of some real vector space which can be triangulated by a finite simplicial complex K whose realization is P = | K |. It is sometimes more convenient to consider a cell-structure C by finitely many convex open cells of various dimensions; each such is the bounded intersection of a finite number of open half-spaces with an affine subspace; the cells in such a structure are disjoint, and the boundary of any cell is a finite union of other cells; such a structure has a triangulation obtained by a “barycentric subdivision” C ′; a vertex is put in each cell, and a set of these form the vertices of a simplex exactly when their corresponding cells are totally ordered by the relation “is in the boundary of”.