Abstract:
A dendron is a continuum (a non-empty connected compact Hausdorff space) in which every two distinct points have a separation point. We prove that if a group G acts on a dendron D by homeomorphisms, then either D contains a G-invariant subset consisting of one or two points, or G contains a free non-commutative subgroup and, furthermore, the action is strongly proximal.
Key words and phrases:
dendron, dendrite, tree, R-tree, pretree, dendritic space, amenability, invariant measure, von Neumann conjecture, Tits alternative, free non-Abelian subgroup, strong proximality.
Citation:
A. V. Malyutin, “Groups acting on dendrons”, Geometry and topology. Part 12, Zap. Nauchn. Sem. POMI, 415, POMI, St. Petersburg, 2013, 62–74; J. Math. Sci. (N. Y.), 212:5 (2016), 558–565
\Bibitem{Mal13}
\by A.~V.~Malyutin
\paper Groups acting on dendrons
\inbook Geometry and topology. Part~12
\serial Zap. Nauchn. Sem. POMI
\yr 2013
\vol 415
\pages 62--74
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl5686}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2016
\vol 212
\issue 5
\pages 558--565
\crossref{https://doi.org/10.1007/s10958-016-2688-2}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84953410388}
Linking options:
https://www.mathnet.ru/eng/znsl5686
https://www.mathnet.ru/eng/znsl/v415/p62
This publication is cited in the following 4 articles:
L. S. Efremova, E. N. Makhrova, “One-dimensional dynamical systems”, Russian Math. Surveys, 76:5 (2021), 821–881
Abdalaoui E.H.E., Naghmouchi I., “Group Action With Finite Orbits on Local Dendrites”, Dynam. Syst., 36:4 (2021), 714–730
Shi E., Ye X., “Equicontinuity of Minimal Sets For Amenable Group Actions on Dendrites”, Dynamics: Topology and Numbers, Contemporary Mathematics, 744, eds. Moree P., Pohl A., Snoha L., Ward T., Amer Mathematical Soc, 2020, 175–180
Glasner E., Megrelishvili M., “Group Actions on Treelike Compact Spaces”, Sci. China-Math., 62:12 (2019), 2447–2462
Citation in format AMSBIB
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