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Free and non-free subgroups of the fundamental group of the Hawaiian Earrings

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EN
The space which is composed by embedding countably many circles in such a way into the plane that their radii are given by a null-sequence and that they all have a common tangent point is called “The Hawaiian Earrings”. The fundamental group of this space is known to be a subgroup of the inverse limit of the finitely generated free groups, and it is known to be not free. Within the recent move of trying to get hands on the algebraic invariants of non-tame (e.g. non-triangulable) spaces this space usually serves as the simplest example in this context. This paper contributes to understanding this group and corresponding phenomena by pointing out that several subgroups that are constructed according to similar schemes partially turn out to be free and not to be free. Amongst them is a countable non-free subgroup, and an uncountable free subgroup that is not contained in two other free subgroups that have recently been found. This group, although free, contains infinitely huge “virtual powers”, i.e. elements of the fundamental group of that kind that are usually used in proofs that this fundamental group is not free, and, although this group contains all homotopy classes of paths that are associated with a single loop of the Hawaiian Earrings, this system of ‘natural generators’ can be proven to be not contained in any free basis of this free group.
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Tom
1
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1
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1-35
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wydano
2003-03-01
online
2003-03-01
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Bibliografia
  • [1] Bogley, W.A. and Sieradski, A.J.: “Universal Path Spaces”, preprint, Oregon State University and University of Oregon, Corvallis and Eugene (Oregon, USA), May 1997
  • [2] Bogley, W.A. and Sieradski, A.J.: “Weighted Combinatorial Group Theory for wild metric complexes”. In “Groups-Korea ’98”, Proceedings of a conference held in Pusan (Korea) 1998, de Gruyter, Berlin, 2000, pp. 53–80
  • [3] Cannon, J.W. and Conner, G.R.: “The Combinatorial Structure of the Hawaiian Earring Group”, Top. Appl. 106, No. 3 (2000), pp. 225–272 http://dx.doi.org/10.1016/S0166-8641(99)00103-0
  • [4] Cannon, J.W. and Conner, G.R.: “The Big Fundamental Group, Big Hawaiian Earrings, and the Big Free Groups” Top. Appl. 106, No. 3 (2000), pp. 273–291 http://dx.doi.org/10.1016/S0166-8641(99)00104-2
  • [5] Dunwoody, M.J.: “Groups acting on protrees”, J. London Math. Soc. (2), Vol. 56, No. 1 (1997), 125–136 http://dx.doi.org/10.1112/S0024610797005322
  • [6] Eda, Katsuya: “Free subgroups of the fundamental group of the Hawaiian Earring”, J. Alg. 219 (1999), pp. 598–605 http://dx.doi.org/10.1006/jabr.1999.7912
  • [7] Eda, Katsuya: “Free σ-products and non-commutatively slender groups”, J. Algebra pp. 243–263
  • [8] Eda, Katsuya: “Free σ-Products and fundamental groups of subspaces of the plane”, Topology Appl. 84 (1998), no. 1–3, pp. 283–306 http://dx.doi.org/10.1016/S0166-8641(97)00105-3
  • [9] Griffiths, H. B.: “The Fundamental group of two spaces with a common point”, Quart. J. Math. Oxford (2), 5 (1954), pp. 175–190 (Correction: ibid. Griffiths, H. B.: “The Fundamental group of two spaces with a common point”, Quart. J. Math. Oxford (2), 6 (1955), pp. 154–155)
  • [10] Griffiths, H. B.: “Infinite Products of semi-groups and local connectivity”, Proc. London Math. Soc. (3), 5 (1956), pp. 455–80
  • [11] Higman, Graham: “Unrestricted free product, and varieties of topological groups”, J. London Math. Soc. (2), 27 (1952), pp. 73–88
  • [12] Kutosh, A.G.: “Zum Zerlegungsproblem der Theorie der freien Produkte”, Rec. Math. Moscow (New Series), 2 (1944), pp. 995–1001
  • [13] Morgan, John and Morrison, Ian: “A van-Kampen Theorem of weak Joins”, Proc. London Math. Soc. (3), 53 (1986), 562–76
  • [14] Sieradski, Alan: “Omega-Groups”, preprint, University of Oregon, Eugene (Oregon, U.S.A.)
  • [15] de Smith, Bart: “The fundamental group of the Hawaiian Earrings is not free”, Internal. J. Algebra. Comput., Vol. 3 (1992), 33–37
  • [16] Zastrow, Andreas: “Construction of an infinitely generated group that is not a free product of surface groups and abelian groups, but which acts freely on an ℝ”, Proc. R. Soc. Edinb., 128A (1998), pp. 433–445
  • [17] Zastrow, Andreas: “The non-abelian Specker-Group is free”, J. Algebra 229, 55–85 (2000) http://dx.doi.org/10.1006/jabr.1999.8261
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bwmeta1.element.doi-10_2478_BF02475662
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