The complexity of maximal cofinitary groups
Author:
Bart Kastermans
Journal:
Proc. Amer. Math. Soc. 137 (2009), 307316
MSC (2000):
Primary 03E47; Secondary 20Bxx
Published electronically:
August 27, 2008
MathSciNet review:
2439455
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Abstract: A cofinitary group is a subgroup of the infinite symmetric group in which each element of the subgroup has at most finitely many fixed points. A maximal cofinitary group is a cofinitary group that is maximal with respect to inclusion. We investigate the possible complexities of maximal cofinitary groups. In particular we show that (1) under the axiom of constructibility there exists a coanalytic maximal cofinitary group and (2) there does not exist an eventually bounded maximal cofinitary group. We also suggest some further directions for investigation.
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 S. A. Adeleke, Embeddings of infinite permutation groups in sharp, highly transitive, and homogeneous groups, Proc. Edinburgh Math. Soc. (2), 31, 1988, pp. 169178. MR 989749 (90e:20003)
 [BSZ00]
 J. Brendle, O. Spinas, and Y. Zhang, Uniformity of the meager ideal and maximal cofinitary groups, J. Algebra, 232, 2000, pp. 209225. MR 1783921 (2001i:03097)
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 P. J. Cameron, Cofinitary permutation groups, Bull. London Math. Soc., 28, 1996, pp. 113140. MR 1367160 (96j:20005)
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 S. Gao and Y. Zhang, Definable sets of generators in maximal cofinitary groups, Advances in Mathematics, 217, 2008, pp. 814832.
 [HSZ01]
 M. Hrušák, J. Steprāns, and Y. Zhang, Cofinitary groups, almost disjoint and dominating families, J. Symbolic Logic, 66, 2001, pp. 12591276. MR 1856740 (2002k:03080)
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 B. Kastermans, Cofinitary Groups and Other Almost Disjoint Families of Reals, Ph.D. Thesis, May 2006, University of Michigan.
 [KSZxx]
 B. Kastermans, J. Steprāns and Y. Zhang, Analytic and coanalytic families of almost disjoint functions, to appear in the Journal of Symbolic Logic.
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 B. Kastermans, Orbits and Isomorphism Types of Maximal Cofinitary Groups, in preparation.
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 A. R. D. Mathias, Happy families, Ann. Math. Logic, 12, 1977, pp. 59111. MR 0491197 (58:10462)
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 A. W. Miller, Infinite combinatorics and definability, Ann. Pure Appl. Logic, 41, 1989, pp. 179203. MR 983001 (90b:03070)
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 J. K. Truss, Embeddings of Infinite Permutation Groups, Proceedings of Groups St. Andrews, 1985, London Math. Soc. Lecture Note Ser., 121, Cambridge Univ. Press, 1986, pp. 335351. MR 896533 (89d:20002)
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 Y. Zhang, Maximal cofinitary groups, Arch. Math. Logic, 39, 2000, pp. 4152. MR 1735183 (2001d:03120)
 [Z03]
 Y. Zhang, Constructing a maximal cofinitary group, Lobachevskii J. Math., 12, 2003, pp. 7381 (electronic). MR 1974545 (2004e:03088)
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Additional Information
Bart Kastermans
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, 480 Lincoln Drive, Madison, Wisconsin 53706
DOI:
http://dx.doi.org/10.1090/S0002993908095269
PII:
S 00029939(08)095269
Received by editor(s):
May 14, 2007
Received by editor(s) in revised form:
January 30, 2008
Published electronically:
August 27, 2008
Additional Notes:
The author was partially supported by Sun YatSen University, Guangzhou, China
Communicated by:
Julia Knight
Article copyright:
© Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
