The action of Thompson's group on a $\mathrm{CAT}(0)$ boundary

Abstract

For a given locally finite $\mathrm{CAT}(0)$) cubical complex $X$ with base vertex $\ast$, we define the profile of a given geodesic ray $c$ issuing from $\ast$ to be the collection of all hyperplanes (in the sense of Sageev) crossed by $c$. We give necessary conditions for a collection of hyperplanes to form the profile of a geodesic ray, and conjecture that these conditions are also sufficient.

We show that profiles in diagram and picture complexes can be expressed naturally as infinite pictures (or diagrams), and use this fact to describe the fixed points at infinity of the actions by Thompson's groups $F,T$, and $V$ on their respective $\mathrm{CAT}(0)$ cubical complexes. In particular, the actions of $T$ and $V$ have no global fixed points. We obtain a partial description of the fixed set of $F$; it consists, at least, of an arc $c$ of Tits length $\pi /2$, and any other fixed points of $F$ must have one particular profile, which we describe. We conjecture that all of the fixed points of $F$ lie on the arc $c$.

Our results are motivated by the problem of determining whether $F$ is amenable.