Every continuous action of a compact group on a uniquely arcwise connected continuum has a fixed point
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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, \(\lambda \)-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.
KeywordsFixed point group action compact group continuum dendrite dendroid lambda dendroid uniquely arcwise connected tree-like
Mathematics Subject ClassificationPrimary 54H25 Secondary 37B45
I am grateful to J. Boroński and R. Mańka for their comments to the first version of this paper.
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