The complexity of maximal cofinitary groups

Kastermans, Bart

doi:10.1090/S0002-9939-08-09526-9

The complexity of maximal cofinitary groups
HTML articles powered by AMS MathViewer

by Bart Kastermans

Proc. Amer. Math. Soc. 137 (2009), 307-316

DOI: https://doi.org/10.1090/S0002-9939-08-09526-9

Published electronically: August 27, 2008

PDF | Request permission

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.

References

S. A. Adeleke, Embeddings of infinite permutation groups in sharp, highly transitive, and homogeneous groups, Proc. Edinburgh Math. Soc. (2) 31 (1988), no. 2, 169–178. MR 989749, DOI 10.1017/S001309150000328X
Jörg Brendle, Otmar Spinas, and Yi Zhang, Uniformity of the meager ideal and maximal cofinitary groups, J. Algebra 232 (2000), no. 1, 209–225. MR 1783921, DOI 10.1006/jabr.2000.8396
Peter J. Cameron, Cofinitary permutation groups, Bull. London Math. Soc. 28 (1996), no. 2, 113–140. MR 1367160, DOI 10.1112/blms/28.2.113
S. Gao and Y. Zhang, Definable sets of generators in maximal cofinitary groups, Advances in Mathematics, 217, 2008, pp. 814–832.
Michael Hrušák, Juris Steprans, and Yi Zhang, Cofinitary groups, almost disjoint and dominating families, J. Symbolic Logic 66 (2001), no. 3, 1259–1276. MR 1856740, DOI 10.2307/2695105
B. Kastermans, Cofinitary Groups and Other Almost Disjoint Families of Reals, Ph.D. Thesis, May 2006, University of Michigan.
B. Kastermans, J. Steprāns and Y. Zhang, Analytic and coanalytic families of almost disjoint functions, to appear in the Journal of Symbolic Logic.
B. Kastermans, Orbits and Isomorphism Types of Maximal Cofinitary Groups, in preparation.
A. R. D. Mathias, Happy families, Ann. Math. Logic 12 (1977), no. 1, 59–111. MR 491197, DOI 10.1016/0003-4843(77)90006-7
Arnold W. Miller, Infinite combinatorics and definability, Ann. Pure Appl. Logic 41 (1989), no. 2, 179–203. MR 983001, DOI 10.1016/0168-0072(89)90013-4
J. K. Truss, Embeddings of infinite permutation groups, Proceedings of groups—St. Andrews 1985, London Math. Soc. Lecture Note Ser., vol. 121, Cambridge Univ. Press, Cambridge, 1986, pp. 335–351. MR 896533
Yi Zhang, Maximal cofinitary groups, Arch. Math. Logic 39 (2000), no. 1, 41–52. MR 1735183, DOI 10.1007/s001530050002
Yi Zhang, Constructing a maximal cofinitary group, Lobachevskii J. Math. 12 (2003), 73–81. MR 1974545