Question:

Are there any well known actions of braid groups on trees? For example is there some action of a braid group $ B_n $ on a $ p $ regular tree for some $ p $ such that the action is transitive on vertices and also transitive on directed edges?

Context:

We often understand a group by its actions. For a finite group or compact group, the theory of its unitary representations is very well behaved. Even for a general reductive algebraic group its representation theory is well behaved.

For finite groups we can also learn things by thinking of the finite group acting as automorphisms of a graph.

I'm not used to working with infinite groups that aren't algebraic groups over some field. Braid groups are an example of infinite groups that aren't algebraic groups over any field. Although there is a theory of unitary representations of Braid groups (e.g. certain values of the Burau representation and Lawrence–Krammer representation) the image of the Braid group is not a closed subgroup of the unitary group and so it is a bit beyond my intuition.

Since the Braid group is an infinite "discrete" group in my intuition it might be related to the automorphisms of an infinite tree.

Cross-posted from MSE https://math.stackexchange.com/questions/4729118/action-of-braid-groups-on-regular-trees

finite degreetree is much stronger than that of a non-elementary action on an arbitrary tree. (And assuming regular of finite degree is even more restrictive.) $\endgroup$Since the Braid group is an infinite "discrete" group in my intuition it might be related to the automorphisms of an infinite tree.Within the class of infinite discrete groups, there is a special subclass FA consisting of groups such that every action on a tree has a global fixed point (see the paper mentioned in the answer below). It is a rich an interesting class, and its complement is a rich and interesting class. So for a general infinite discrete group, there is no particular reason to expect it to lie in FA or its complement. $\endgroup$