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Homogeneity groups of ends of open 3-manifolds

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  • For every finitely generated abelian group G, we construct an irreducible open 3-manifold M[subscript G] whose end set is homeomorphic to a Cantor set and whose homogeneity group is isomorphic to G. The end homogeneity group is the group of self-homeomorphisms of the end set that extend to homeomorphisms of the 3-manifold. The techniques involve computing the embedding homogeneity groups of carefully constructed Antoine-type Cantor sets made up of rigid pieces. In addition, a generalization of an Antoine Cantor set using infinite chains is needed to construct an example with integer homogeneity group. Results about the local genus of points in Cantor sets and about the geometric index are also used.
  • This is the publisher’s final pdf. The published article is copyrighted by the Mathematical Sciences Publishers and can be found at: http://msp.org/pjm/2014/269-1/index.xhtml.
  • Keywords: Manifold end, Homogeneity group, Defining sequence, Geometric index, Cantor set, Abelian group, Rigidity, Open 3-manifold
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  • Garity, D. J., & Repovš, D. (2014). Homogeneity groups of ends of open 3-manifolds. Pacific Journal of Mathematics, 269(1), 99-112. doi:10.2140/pjm.2014.269.99
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  • 269
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  • 1
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  • Garity was supported in part by NSF grants DMS-0852030 and DMS-1005906. Both authors were supported in part by the Slovenian Research Agency grant BI-US/11-12-023 and BI-US/13-14-027. Repovš was supported in part by Slovenian Research Agency grants P1-0292-0101, J1-2057-0101, and J1-4144-0101.
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