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