The geometry and fundamental groups of solenoid complements

Conner, Gregory R. (author), Meilstrup, Mark H. (author), Repovš, Dušan (author)

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 nonhomeomorphic complements.

• Language: English
• Keywords: solenoid, 3-manifold, inverse limit, embedding, fundamental group, knot complement, braid, Jaco-Shalen-Johannson decomposition, Mostow-Prasad rigidity
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Year: 2015
• Number of pages: 20 str.
• Numbering: Vol. 24, iss. 14
• PID: 20.500.12556/RUL-111223
• UDC: 515.162
• ISSN on article: 0218-2165
• DOI: 10.1142/S0218216515500698
• COBISS.SI-ID: 17540697
• Publication date in RUL: 26.09.2019

Record is a part of a journal

• Title: Journal of knot theory and its ramifications
• Shortened title: J. knot theory ramif.
• Publisher: World Scientific
• ISSN: 0218-2165
• COBISS.SI-ID: 3996249