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Highly-transitive actions of surface groups

Author: Daniel Kitroser
Journal: Proc. Amer. Math. Soc. 140 (2012), 3365-3375
MSC (2010): Primary 20B22, 20B35
Published electronically: April 24, 2012
MathSciNet review: 2929006
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Abstract: A group action is said to be highly-transitive if it is $ k$-transitive for every $ k\geq 1$. The main result of this thesis is the following:

Main Theorem. The fundamental group of a closed, orientable surface of genus $ > 1$ admits a faithful, highly-transitive action on a countably infinite set.

From a topological point of view, finding a faithful, highly-transitive action of a surface group is equivalent to finding an embedding of the surface group into $ \operatorname {sym}{\mathbb{Z}}$ with a dense image. In this topological setting, we use methods that were originally developed for densely embedding surface groups in locally compact groups.

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Daniel Kitroser
Affiliation: Department of Mathematics, Ben-Gurion University of The Negev, Be’er Sheva, Israel

Received by editor(s): December 15, 2010
Received by editor(s) in revised form: April 11, 2011
Published electronically: April 24, 2012
Additional Notes: The author was partially supported by ISF grant 888/07
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2012 American Mathematical Society