TITLE: Actions of Thompson group F
QUESTION [9 upvotes]: Does anybody know the actions of Thompson group F which are not conjugate to the standard one?
Motivation is to find actions such that the Schreier graph of the action does not contain a binary tree.
I've decided to ask a separate question on the motivation here

REPLY [5 votes]: $F$ is bi-orderable, so if you care about actions by homeomorphisms on the line, you can pick up a bi-ordering and produce a dynamical realization: a faithful almost free action ( see Proposition 3.4 and Example 3.5 in this paper of Navas http://arxiv.org/abs/0710.2466  for a "non-standard" action of $F$). For the discrete case, the proof of Proposition 1.8 in the same paper might be an inspiration... All the bi-orderings of $F$ were described by Navas and Rivas here http://arxiv.org/abs/0808.1688 . (As a side remark, in one dimensional dynamics one usually consider two actions equivalent if they are semi-conjugate, rather than conjugate) 
(this was meant to be just a comment, but don't have enough points...:)