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arXiv · 1709.00852

Every continuous action of a compact group on a uniquely arcwise connected continuum has a fixed point

Benjamin Vejnar
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Abstract

We are dealing with the question whether every group or semigroup action (with some additional property) on a continuum (with some additional property) has a fixed point. One of such results was given in 2009 by Shi and Sun. They proved that every nilpotent group action on a uniquely arcwise connected continuum has a fixed point. We are seeking for this type of results with e.g. commutative, compact or torsion groups and semigroups acting on dendrites, dendroids, $λ$-dendroids and uniquely arcwise connected continua. We prove that every continuous action of a compact or torsion group on a uniquely arcwise connected continuum has a fixed point. We also prove that every continuous action of a compact and commutative semigroup on a uniquely arcwise connected continuum or on a tree-like continuum has a fixed point.

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Benjamin Vejnar. 2017-12-22. Every continuous action of a compact group on a uniquely arcwise connected continuum has a fixed point. https://arxiv.org/abs/1709.00852

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