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UNIVERSAL MINIMAL FLOWS OF GENERALIZED WAŻEWSKI DENDRITES

Published online by Cambridge University Press:  21 December 2018

ALEKSANDRA KWIATKOWSKA
ALEKSANDRA KWIATKOWSKA*
Affiliation:
INSTITUT FÜR MATHEMATISCHE LOGIK UND GRUNDLAGENFORSCHUNGUNIVERSITÄT MÜNSTEREINSTEINSTRASSE 62 48149 MÜNSTER, GERMANY and INSTYTUT MATEMATYCZNYUNIWERSYTET WROCŁAWSKIPL. GRUNWALDZKI 2/4 50-384 WROCŁAW, POLANDE-mail: [email protected]
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Abstract

We study universal minimal flows of the homeomorphism groups of generalized Ważewski dendrites WP, $P \subseteq \left\{ {3,4, \ldots ,\omega } \right\}$. If P is finite, we prove that the universal minimal flow of the homeomorphism group H (WP) is metrizable and we compute it explicitly. This answers a question of Duchesne. If P is infinite, we show that the universal minimal flow of H (WP) is not metrizable. This provides examples of topological groups which are Roelcke precompact and have a nonmetrizable universal minimal flow with a comeager orbit.

Keywords

05D1037B0554F1503C98universal minimal flowshomeomorphism groups of Ważewski dendritesFraïssé limits
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 4 , December 2018 , pp. 1618 - 1632
Copyright
Copyright © The Association for Symbolic Logic 2018 

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References

REFERENCES

Adeleke, S. A. and Neumann, P. M., Relations related to betweenness: Their structure and automorphisms. Memoirs of the American Mathematical Society, vol. 131 (1998), no. 623, viii+125 pp.CrossRefGoogle Scholar
Bartošová, D. and Kwiatkowska, A., The universal minimal flow of the homeomorphism group of the Lelek fan. Transactions of the American Mathematical Society, to appear, arXiv:1706.09154.Google Scholar
Ben Yaacov, I., Melleray, J., and Tsankov, T., Metrizable universal minimal flows of Polish groups have a comeager orbit. Geometric and Functional Analysis, vol. 27 (2017), no. 1, pp. 67–77.CrossRefGoogle Scholar
Bodirsky, M. and Piguet, D., Finite trees are Ramsey under topological embeddings, preprint, 2010, arXiv:1002.1557.Google Scholar
Bodirsky, M., Pinsker, M., and Tsankov, T., Decidability of definability, this Journal, vol. 78 (2013), no. 4, pp. 1036–1054.Google Scholar
Charatonik, W. J. and Dilks, A. M., On self-homeomorphic spaces. Topology and its Applications, vol. 55 (1994), no. 3, pp. 215–238.CrossRefGoogle Scholar
Deuber, W., A generalization of Ramsey’s theorem for regular trees. Journal of Combinatorial Theory, Series B, vol. 18 (1975), pp. 18–23.CrossRefGoogle Scholar
Duchesne, B., Topological properties of Ważewski dendrite groups, preprint, 2017.Google Scholar
Duchesne, B. and Monod, N., Structural properties of dendrite groups. Transactions of the American Mathematical Society, to appear, arXiv:1610.08488.Google Scholar
Evans, D., Homogeneous structures, ω-categoricity and amalgamation constructions, Notes on a Minicourse given at HIM, Bonn, September 2013. Available at https://www.him.uni-bonn.de/programs/past-programs/past-trimester-programs/universality-and-homogeneity/mini-courses/.Google Scholar
Evans, D., Hubička, J., and Nešetřil, J., Automorphism groups and Ramsey properties of sparse graphs, preprint, 2017, arXiv:1801.01165.Google Scholar
Hrushovski, E., A stable ${\aleph _0}$-categorical pseudoplane, unpublished notes, 1988.Google Scholar
Jasiński, J., Ramsey degrees of boron tree structures. Combinatorica, vol. 33 (2013), no. 1, pp. 23–44.CrossRefGoogle Scholar
Kechris, A. S., Pestov, V. G., and Todorcevic, S., Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups. Geometric and Functional Analysis, vol. 15 (2005), no. 1, pp. 106–189.CrossRefGoogle Scholar
Melleray, J., Nguyen Van Thé, L., and Tsankov, T., Polish groups with metrizable universal minimal flows. International Mathematics Research Notices, vol. 2016 (2016), no. 5, pp. 1285–1307.CrossRefGoogle Scholar
Nešetřil, J. and Rödl, V., Partitions of finite relational and set systems. Journal of Combinatorial Theory, Series A, vol. 22 (1977), no. 3, pp. 289–312.CrossRefGoogle Scholar
Nguyen Van Thé, L., More on the Kechris-Pestov-Todorcevic correspondence: Precompact expansions. Fundamenta Mathematicae, vol. 222 (2013), no. 1, pp. 19–47.CrossRefGoogle Scholar
Sokić, M., Semilattices and the Ramsey property, this Journal, vol. 80 (2015), no. 4, pp. 1236–1259.Google Scholar
Sokić, M., Ramsey property of finite posets II. Order, vol. 29 (2012), no. 1, pp. 31–47.CrossRefGoogle Scholar
Solecki, S., Abstract approach to Ramsey theory and Ramsey theorems for finite trees, Asymptotic Geometric Analysis, Fields Institute Communications (Ludwig, M., Milman, V. D., Pestov, W., and Tomczak-Jaegermann, N., editors), Springer, New York, 2013, pp. 313–340.CrossRefGoogle Scholar
Tsankov, T., Unitary representations of oligomorphic groups. Geometric and Functional Analysis, vol. 22 (2012), no. 2, pp. 528–555.CrossRefGoogle Scholar
Zucker, A., Topological dynamics of automorphism groups, ultrafilter combinatorics, and the Generic Point Problem. Transactions of the American Mathematical Society, vol. 368 (2016), no. 9, pp. 6715–6740.CrossRefGoogle Scholar
Zucker, A., New directions in the abstract topological dynamics of Polish groups, Ph.D. thesis, Carnegie Mellon University, April 2018.Google Scholar