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I have seen many examples where a finitely generated infinite group acts properly/freely by isometry on finite dimensional CAT(0) cube complexes. Examples of such groups are discussed in many articles.

My question is

What are the examples of infinite, finitely generated groups that acts properly (or freely) on infinite dimensional CAT(0) cube complex?

I also come across the statement in the paper by Roller and Niblo where finite dimension condition is not required (or I don't know if they need it somehow).

Theorem A group acts fixed-point-freely on a CAT(0) cube complex if and only if it contains a codimension-one subgroup.

Do these groups act properly on infinite dimensional CAT(0) cube complex?

I thought my search is over after going through the MO post when I came across the example of Thompson groups T and V. However, these groups are not finitely generated.

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    $\begingroup$ Eh? $T$ and $V$ are in fact finitely generated (and finitely presented, and type $F_\infty$). I think they are a perfect example of the sort of thing you're asking about. $\endgroup$
    – Matt Zaremsky
    Commented Nov 25, 2023 at 22:06
  • $\begingroup$ Thanks for both the answers! I didn't know that they are finitely generated. $\endgroup$
    – bishop1989
    Commented Nov 25, 2023 at 23:20
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    $\begingroup$ Worth mentioning one little fact where the complex need not be finite dimensional: If a group acts properly on a CAT(0) cube complex then the group has the Haagerup property. This is the main thing I'm aware of that "still works" without finite dimensionality. $\endgroup$
    – Matt Zaremsky
    Commented Nov 26, 2023 at 13:55
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    $\begingroup$ This is called "Property PW" in some papers. I studied/surveyed it here. $\endgroup$
    – YCor
    Commented Nov 26, 2023 at 18:37

1 Answer 1

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Natural examples of finitely generated groups acting properly on median graphs of infinite cubical dimension (or, if you prefer, on infinite-dimensional CAT(0) cube complexes) include:

  • Thompson's groups $F$, $T$, and $V$. There are also many other Thompson-like groups.
  • Groups defined by infinite small cancellation presentations.
  • Grigorchuk groups.
  • Some lamplighter groups, e.g. lamplighters over free groups.
  • Some tubular groups.
  • Some diagram groups.

About groups containing codimension-one subgroups, there is no reason for your fixed-point free action to be proper. For instance, every finitely generated group with an infinite abelianisation contains a codimension-one subgroup but it may not act properly on a median graph. A stupid example could be a product $G \times \mathbb{Z}$ where $G$ has Property (T). Or a free product of groups with (T) acts non-trivially on a tree but does not act properly on a median graph.

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