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ON THE AUTOMORPHISM GROUP OF THE UNIVERSAL HOMOGENEOUS MEET-TREE

Part of: Ordered sets Connections with other structures, applications Set theory Model theory Permutation groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  01 February 2021

ITAY KAPLAN ,
TOMASZ RZEPECKI  and
DAOUD SINIORA
ITAY KAPLAN
Affiliation:
EINSTEIN INSTITUTE OF MATHEMATICS HEBREW UNIVERSITY OF JERUSALEM 91904, JERUSALEM, ISRAELE-mail: kaplan@math.huji.ac.il
TOMASZ RZEPECKI
Affiliation:
EINSTEIN INSTITUTE OF MATHEMATICS HEBREW UNIVERSITY OF JERUSALEM 91904, JERUSALEM, ISRAEL and INSTYTUT MATEMATYCZNY UNIWERSYTET WROCŁAWSKI PL. GRUNWALDZKI 2/4 50-384 WROCŁAW, POLANDE-mail: tomasz.rzepecki@math.uni.wroc.pl
DAOUD SINIORA
Affiliation:
INDEPENDENT SCHOLAR E-mail: daoud.siniora@gmail.com
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Abstract

We show that the countable universal homogeneous meet-tree has a generic automorphism, but it does not have a generic pair of automorphisms.

Keywords

Fraïssé limitgeneric automorphismtreePolish groupample generics

MSC classification

Primary: 03C15: Denumerable structures
Secondary: 03E15: Descriptive set theory 06A12: Semilattices 20E08: Groups acting on trees 54H11: Topological groups 20B27: Infinite automorphism groups
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 4 , December 2021 , pp. 1508 - 1540
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Copyright
© Association for Symbolic Logic 2021

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References

Bodirsky, M., Bradley-Williams, D., Pinsker, M., and Pongrácz, A., The universal homogeneous binary tree . Journal of Logic and Computation, vol. 28 (2018), no. 1, pp. 133–163.10.1093/logcom/exx043CrossRefGoogle Scholar
Droste, M., Holland, W. C., and Macpherson, H. D., Automorphism groups of infinite semilinear orders (II) . Proceedings of the London Mathematical Society, vol. s3-58 (1989), no. 3, pp. 479–494.10.1112/plms/s3-58.3.479CrossRefGoogle Scholar
Duchesne, B., Topological properties of Ważewski dendrite groups . Journal de l’École polytechnique—Mathématiques, vol. 7 (2020), pp. 431–477.10.5802/jep.121CrossRefGoogle Scholar
Hodges, W., A Shorter Model Theory, Cambridge University Press, New York, NY, 1997.Google Scholar
Ivanov, A. A., Generic expansions of $\omega$ -categorical structures and semantics of generalized quantifiers. this Journal, vol. 64 (1999), no. 2, pp. 775–789.Google Scholar
Kaplan, I. and Shelah, S., A dependent theory with few indiscernibles . Israel Journal of Mathematics, vol. 202 (2014), no. 1, pp. 59–103.10.1007/s11856-014-1067-2CrossRefGoogle Scholar
Kaplan, I. and Shelah, S., Examples in dependent theories , this Journal, vol. 79 (2014), no. 2, pp.585–619.Google Scholar
Kechris, A. S., Pestov, V. G., and Todorcevic, S., Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups. Geometric & Functional Analysis, vol. 15 (2005), no. 1, pp. 106–189.10.1007/s00039-005-0503-1CrossRefGoogle Scholar
Kechris, A. S. and Rosendal, C., Turbulence, amalgamation, and generic automorphisms of homogeneous structures . Proceedings of the London Mathematical Society, vol. 94 (2007), no. 2, pp. 302–350.10.1112/plms/pdl007CrossRefGoogle Scholar
Kuske, D. and Truss, J. K., Generic automorphisms of the universal partial order . Proceedings of the American Mathematical Society, vol. 129 (2001), no. 7, pp. 1939–1948.CrossRefGoogle Scholar
Kwiatkowska, A., The group of homeomorphisms of the cantor set has ample generics . Bulletin of the London Mathematical Society, vol. 44 (2012), no. 6, pp. 1132–1146.10.1112/blms/bds039CrossRefGoogle Scholar
Kwiatkowska, A. and Malicki, M., Ordered structures and large conjugacy classes . Journal of Algebra, vol. 557 (2020), pp. 67–96.10.1016/j.jalgebra.2020.03.021CrossRefGoogle Scholar
Macpherson, D., A survey of homogeneous structures . Discrete Mathematics, vol. 311 (2011), no. 15, pp. 1599–1634.10.1016/j.disc.2011.01.024CrossRefGoogle Scholar
Simon, P., A Guide to NIP Theories , Lecture Notes in Logic, vol. 44, Association for Symbolic Logic and Cambridge Scientific, Chicago, IL and Cambridge, 2015.Google Scholar
Simon, P., NIP omega-categorical structures: The rank 1 case, arXiv preprint, 2018, arXiv:1807.07102.Google Scholar
Siniora, D. N., Automorphism groups of homogeneous structures , Ph.D. thesis, University of Leeds, 2017.Google Scholar
Truss, J. K., Generic automorphisms of homogeneous structures . Proceedings of the American Mathematical Society, vol. s3-65 (1992), no. 1, pp. 121–141.10.1112/plms/s3-65.1.121CrossRefGoogle Scholar
Truss, J. K., On notions of genericity and mutual genericity , this Journal, vol. 72 (2007), no. 3, pp. 755–766.Google Scholar