# Groups, Geometry, and Dynamics

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**Volume 2, Issue 2, 2008, pp. 185–222**

**DOI: 10.4171/GGD/36**

Published online: 2008-06-30

The action of Thompson's group on a CAT(0) boundary

Daniel Farley^{[1]}(1) Miami University, Oxford, United States

For a given locally finite CAT(0) cubical complex `X` with base
vertex ∗, we define the *profile* of a given geodesic ray
`c` issuing from ∗ 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 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 `π`/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.

*Keywords: *Amenability, CAT(0) cubical complex, Thompson's group, diagram group, space at infinity

Farley Daniel: The action of Thompson's group on a CAT(0) boundary. *Groups Geom. Dyn.* 2 (2008), 185-222. doi: 10.4171/GGD/36