Free groups as end homogeneity groups of 3-manifolds

Garity, Dennis (author), Repovš, Dušan (author)

For every finitely generated free group ▫$F$▫, we construct an irreducible open 3-manifold ▫$M_F$▫ whose end set is homeomorphic to a Cantor set, and with the end homogeneity group of ▫$M_F$▫ isomorphic to ▫$F$▫. The end homogeneity group is the group of all self-homeomorphisms of the end set that extend to homeomorphisms of the entire 3-manifold. This extends an earlier result that constructs, for each finitely generated abelian group ▫$G$▫, an irreducible open 3-manifold ▫$M_G$▫ with end homogeneity group ▫$G$▫. The method used in the proof of our main result also shows that if ▫$G$▫ is a group with a Cayley graph in ▫$\mathbb{R}^3$▫ such that the graph automorphisms have certain nice extension properties, then there is an irreducible open 3-manifold ▫$M_G$▫ with end homogeneity group ▫$G$▫.

• Language: English
• Keywords: open 3-manifold, rigidity, manifold end, geometric index, Cantor set, homogeneity group, abelian group, defining sequence
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Publication version: Author Accepted Manuscript
• Year: 2022
• Number of pages: Str. 113-130
• Numbering: Vol. 38, iss. 1
• PID: 20.500.12556/RUL-135295
• UDC: 515.122
• ISSN on article: 0213-2230
• DOI: 10.4171/RMI/1273
• COBISS.SI-ID: 69890563
• Publication date in RUL: 07.03.2022

Record is a part of a journal

• Title: Revista matemática iberoamericana
• Shortened title: Rev. mat. iberoam.
• Publisher: Consejo Superior de Investigaciones Científicas, Real Sociedad Matemática Española, Antonio Córdoba
• ISSN: 0213-2230
• COBISS.SI-ID: 26288128