In this talk, I will present recent results about the poset of hyperbolic structures on Thompson’s group $F$. While the global structure of this poset is as simple as one would expect, the local structure turns out to be incredibly rich, in stark contrast with the situation for the $T$ and $V$ counterparts.

I will focus on the subposet of quasi-parabolic hyperbolic structures, which contains uncountably many \emph{lamplike} structures, called so as they can be described combinatorially in terms of certain hyperbolic structures on related lamplighter groups. On the other hand, there are also many non-lamplike structures, showing the vastness and complexity of this poset. Lastly, I will talk about how these actions can be extended to more general Thompson’s groups $F_n$ for $n\ge 2$. This is joint work with Francesco Fournier-Facio and Matthew C.B.Zaremsky.