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Transactions of the American Mathematical Society

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

   
 
 

 

Automorphism groups of finite topological rank


Authors: Itay Kaplan and Pierre Simon
Journal: Trans. Amer. Math. Soc. 372 (2019), 2011-2043
MSC (2010): Primary 20B27, 03C15
DOI: https://doi.org/10.1090/tran/7674
Published electronically: April 12, 2019
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Abstract: We offer a criterion for showing that the automorphism group of an ultrahomogeneous structure is topologically 2-generated and even has a cyclically dense conjugacy class. We then show how finite topological rank of the automorphism group of an $ \omega $-categorical structure can go down to reducts. Together, those results prove that a large number of $ \omega $-categorical structures that appear in the literature have an automorphism group of finite topological rank. In fact, we are not aware of any $ \omega $-categorical structure to which they do not apply (assuming the automorphism group has no compact quotients). We end with a few questions and conjectures.


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Additional Information

Itay Kaplan
Affiliation: Institute of Mathematics Hebrew University (The Edmond J. Safra Campus, Giv’at Ram) Jerusalem 91904, Israel

Pierre Simon
Affiliation: Mathematics Department, University of California, Berkeley, Evans Hall, Berkeley, California, 94720-3840

DOI: https://doi.org/10.1090/tran/7674
Received by editor(s): February 6, 2018
Received by editor(s) in revised form: July 18, 2018
Published electronically: April 12, 2019
Additional Notes: The first author would like to thank the Israel Science Foundation for partial support of this research (Grants no. 1533/14 and 1254/18).
The second author was partially supported by NSF (grant DMS 1665491), and the Sloan foundation.
Article copyright: © Copyright 2019 Itay Kaplan and Pierre Simon