Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Some applications of tree-limits to groups. I
HTML articles powered by AMS MathViewer

by Kenneth Hickin PDF
Trans. Amer. Math. Soc. 305 (1988), 797-839 Request permission

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.
References
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 03C30, 20B22, 20E18
  • Retrieve articles in all journals with MSC: 03C30, 20B22, 20E18
Additional Information
  • © Copyright 1988 American Mathematical Society
  • 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