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Archipelago groups


Authors: Gregory R. Conner, Wolfram Hojka and Mark Meilstrup
Journal: Proc. Amer. Math. Soc. 143 (2015), 4973-4988
MSC (2010): Primary 55Q20, 20E06; Secondary 57M30, 57M05, 20F05
DOI: https://doi.org/10.1090/proc/12609
Published electronically: June 5, 2015
MathSciNet review: 3391054
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Abstract: The classical archipelago is a non-contractible subset of $ \mathbb{R}^3$ which is homeomorphic to a disk except at one non-manifold point. Its fundamental group, $ \mathscr {A}$, is the quotient of the topologist's product of $ \mathbb{Z}$, the fundamental group of the shrinking wedge of countably many copies of the circle (the Hawaiian earring), modulo the corresponding free product. We show $ \mathscr {A}$ is locally free, not indicable, and has the rationals both as a subgroup and a quotient group. Replacing $ \mathbb{Z}$ with arbitrary groups yields the notion of archipelago groups.

Surprisingly, every archipelago of countable groups is isomorphic to either $ \mathscr {A}(\mathbb{Z})$ or $ \mathscr {A}(\mathbb{Z}_2)$, the cases where the archipelago is built from circles or projective planes respectively. We conjecture that these two groups are isomorphic and prove that for large enough cardinalities of $ G_i$, $ \mathscr {A}(G_i)$ is not isomorphic to either.


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Additional Information

Gregory R. Conner
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email: conner@math.byu.edu

Wolfram Hojka
Affiliation: Institute for Analysis and Scientific Computation, Technische Universität Wien, Vienna, Austria
Email: w.hojka@gmail.com

Mark Meilstrup
Affiliation: Mathematics Department, Southern Utah University, Cedar City, Utah 84720
Email: mark.meilstrup@gmail.com

DOI: https://doi.org/10.1090/proc/12609
Keywords: Archipelago, topologist's product, mapping cone, wedge, infinite word
Received by editor(s): November 6, 2013
Received by editor(s) in revised form: August 15, 2014
Published electronically: June 5, 2015
Additional Notes: This work was supported by the Simons Foundation Grant 246221 and by the Austrian Science Foundation FWF project S9612.
Communicated by: Kevin Whyte
Article copyright: © Copyright 2015 American Mathematical Society