The geometry and fundamental groups of solenoid complements

Abstract

Abstract. A solenoid is an inverse limit of circles. When a solenoid is embedded in three space, its complement is an open three manifold. We discuss the geometry and fundamental groups of such manifolds, and show that the complements of different solenoids (arising from different inverse limits) have different fundamental groups. Embeddings of the same solenoid can give different groups; in particular, the nicest embeddings are unknotted at each level, and give an Abelian fundamental group, while other embeddings have non-Abelian groups. We show using geometry that every solenoid has uncountably many embeddings with non-homeomorphic complements.In this paper we study 3-manifolds which are complements of solenoids in S 3 . This theory is a natural extension of the study of knot complements in S 3 ; many of the tools that we use are the same as those used in knot theory and braid theory.We will mainly be concerned with studying the geometry and fundamental groups of 3-manifolds which are solenoid complements. We review basic information about solenoids in section 1. In section 2 we discuss the calculation of the fundamental group of solenoid complements. In section 3 we show that every solenoid has an embedding in S 3 so that the complementary 3-manifold has an Abelian fundamental group, which is in fact a subgroup of Q (Theorem 3.5). In section 4 we show that each solenoid has an embedding whose complement has a non-Abelian fundamental group (Theorem 4.3). In section 5 we take a more geometric approach, and show that each solenoid admits uncountably many embeddings in S 3 with non-homeomorphic complements (Theorem 5.4). We achieve this by showing that these complements have distinct geometries using JSJ theory, and thus by Mostow-Prasad rigidity are distinct manifolds.

Definable (co)homology, pro-torus rigidity, and (co)homological classification

“…As we have seen, the study of solenoid complements traces back to the work of Borsuk and Eilenberg from the 1930s [7]. The geometry of solenoid complements has been studied more recently in the context of knot theory in [12,33].…”

The closure of two-sided multiplications on C*-algebras and phantom line bundles

“…In private correspondence, Mladen Bestvina informed us that we can find phantom bundles even over some open subset of R 3 , and referred us to [6]. We outline the construction of such a subset.…”

“…Proof. In [6], a construction is given of dense open sets U in the 3-sphere S 3 with fundamental groups π 1 (U ) that are large subgroups of Q. Given a sequence n i of natural numbers n i > 1, π 1 (U ) can be {p/q ∈ Q : p ∈ Z, q = k i=1 n i for some k}.…”

Knots and solenoids that cannot be attractors of self-homeomorphisms of $\mathbb{R}^3$

“…Our interest in this stems from its applications to the realizability problem stated above: we shall prove that toroidal attractors must have finite genus and, using this, we will be able to construct (uncountably) many different examples of toroidal sets that cannot be realized as attractors because they have infinite genus; namely some wild knots that are expressed as an infinite connected sum of non-trivial knots and also knotted solenoids. Knotted solenoids were studied by Conner, Meilstrup and Repovs in [7].…”