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

TL;DR

This paper proves that every continuous action of a compact or torsion group, and certain semigroups, on a uniquely arcwise connected continuum or tree-like continuum has a fixed point, extending previous fixed point results.

Contribution

It establishes fixed point properties for actions of compact, torsion, and commutative semigroups and groups on specific continua, generalizing earlier results.

Findings

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

Every continuous action of a compact and commutative semigroup on a uniquely arcwise connected continuum has a fixed point.

Fixed point results extend to actions on dendrites, dendroids, and tree-like continua.

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, $\lambda$-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.