William B. Dickson

Branched 1-Manifolds and Presentations of Solenoids

September 1998.

Supervisor: Chris Athorne

Abstract:

The aim of the project was to study the presentations of solenoids by branched 1-manifolds. We begin by studying two properties of branched 1-manifolds which effect the presentation of solenoids, orientability and recurrence. Solenoids are shown to come in two varieties, those presented by orientable branched 1-manifolds and those presented by non-orientable branched 1-manifolds. Two methods of determining whether or not a branched 1-manifold is orientable are given. Recurrence is shown to be a necessary and sufficient condition for a branched 1-manifold to present a solenoid. We then show how the question of whether or not a branched 1-manifold is recurrent can be converted into a question in graph theory for which there exist efficient algorithms.

Next we consider two special types of presentation, elementary presentation and (p,q)-block presentation, which allow us to extract algebraic invariants for the equivalence of solenoids. A slightly stronger version of a result of Williams [Topology 6 (1967) 473-487; MathSciNet Review] is obtained which states that any solenoid with a fixed point is equivalent to one presented by an elementary presentation. The proof is constructive and gives a method for finding an elementary presentation given a presentation of a solenoid with a fixed point. Williams has shown [ibid.] that there is a complete invariant for the equivalence of solenoids given by an elementary presentation in terms of the shift equivalence of endomorphisms of a free group. We prove a result which shows that every solenoid can be given by a coutably infinite number of equivalent (p,q)-block presentations. The (p,q)-block presentations again allow us to find invarients for the equivalence of solenoids in terms of the shift equivalence of endomorphisms of a free group.

Finally we consider further invariants for the equivalence of solenoids which are derived from the endomorphism invariants. First we examine the invariants which arise upon abelianizing the free group in question. Second we introduce new invariants which reflect some of the non-abelian character of these endomorphisms. These new non-abelian invariants are then used to solve a problem posed by Williams [ibid.]