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Some applications of tree-limits to groups. I


Author: Kenneth Hickin
Journal: Trans. Amer. Math. Soc. 305 (1988), 797-839
MSC: Primary 03C30; Secondary 20B22, 20E18
DOI: https://doi.org/10.1090/S0002-9947-1988-0924778-2
MathSciNet review: 924778
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Abstract: Sharper applications to group theory are given of an elegant construction -- the "tree-limit"--which S. Shelah circulated as a preprint in 1977 and used to obtain $ \infty $-$ \omega $-enlargements to power $ {2^\omega }$ of certain countable homogeneous groups and skew fields. In this paper we enlarge the class of groups to which this construction can be interestingly applied and we obtain permutation representations of countable degree of the tree-limit groups; we obtain uncountable subgroup-incomparability for enlargements of countable existentially closed groups and even in nonhomogeneous cases we obtain the very strong "archetypal direct limit property" (which implies $ \infty $-$ \omega $-equivalence (see (1.0)) of the permutation representations). We are able to control the permutation representations which get stretched by the tree-limit by varying the point-stabilizer subgroups (see (5.5)). In particular we can archetypally stretch in $ {2^\omega }$ subgroup-incomparable ways any homogeneous permutation representation of a countable locally finite group in which every finite subgroup has infinitely many regular orbits (Theorem 4). We discuss cases where tree-limits are subgroups of inverse limits.


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  • [1] R. Baer, Local endlich-auflobare gruppen mit endlichen Sylowuntergruppen, J. Reine Angew. Math. 239 (1969), 109-44. MR 0258962 (41:3607)
  • [2] V. Faber, Large abelian subgroups of some infinite groups, Rocky Mountain J. Math. 1 (1971), 677-685. MR 0285617 (44:2835)
  • [3] D. Giorgetta and S. Shelah, Existentially closed structures in the power of the continuum, Ann. Pure Appl. Logic 26 (1984), 123-148. MR 739576 (86e:03035)
  • [4] K. Hickin, Relatively homogeneous locally finite permutation groups, Math. Z. 194 (1987), 495-504. MR 881707 (88b:20044)
  • [5] -, Complete universal locally finite groups, Trans. Amer. Math. Soc. 239 (1978), 213-227. MR 0480750 (58:902)
  • [6] K. Hickin and A. Macintyre, A. C. groups: embeddings and centralizers, Word Problems II, North-Holland, Amsterdam, 1980, pp. 141-155. MR 579943 (82c:20065)
  • [7] O. Kegel, Examples of highly transitive permutation groups, Rend. Sem. Mat. Univ. Padova 63 (1980), 295-300. MR 605800 (82f:20011)
  • [8] F. Leinen, Existentially closed locally finite $ p$-groups, J. Algebra 103 (1986), 160-183. MR 860695 (88b:20004)
  • [9] -, Existentially closed groups in locally finite group classes, Comm. Algebra 13 (1985), 1991-2024. MR 795488 (87b:20042)
  • [10] B. Maier, Existenziell abgeschloshene lokal endliche $ p$-gruppen, Arch. Math. 37 (1981), 113-128. MR 640796 (83e:20005)
  • [11] S. Shelah, Existentially closed models in continuum, Proc. Logic Workshop, Berlin (July 1977), preprint. MR 528419 (80j:03047)
  • [12] S. Shelah and M. Ziegler, Algebraically closed groups of large cardinality, J. Symbolic Logic 44 (1979). 130-140. MR 550381 (80j:03048)
  • [13] -, Uncountable constructions for Boolean algebras, e.c. groups, and Banach spaces, Israel J. Math. 51 (1985), 273. MR 804487 (87d:03096)
  • [14] S. Thomas, Complete existentially closed locally finite groups, Arch. Math. 44 (1985), 97-109. MR 780255 (86h:03064)
  • [15] Jon Hall, Infinite alternating groups as finitary linear transformation groups, preprint. MR 971138 (90a:20079)
  • [16] R. Lyndon and P. Schupp, Combinatorial group theory, Springer-Verlag, 1977. MR 0577064 (58:28182)

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DOI: https://doi.org/10.1090/S0002-9947-1988-0924778-2
Article copyright: © Copyright 1988 American Mathematical Society

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