# Groups acting on trees

Assume that $$X$$ is a tree such that every vertex has infinite degree, and a discrete group $$G$$ acts on this tree properly (with finite stabilizers) and transitively.

Is it true that $$G$$ contains a non-abelian free subgroup?

• Properly implies finite stabilizers
– YCor
Mar 1 '19 at 23:09

Yes.

You're assuming more than what's necessary.

For an isometric group action on a Gromov-hyperbolic space, you have 5 possibilities (Gromov's classification):

• (a) bounded orbits
• (b) horocyclic (fixes a unique point at infinity, no loxodromic element; preserves "horospheres")
• (c) axial (preserves an axis, on which some element acts loxodromically)
• (d) focal (fixes a unique point at infinity, existence of a loxodromic element)
• (e) general type (= other): implies the existence of a non-abelian free subgroup acting metrically properly.

(a), (b), (c) are clearly ruled out in your setting (transitivity, and valency; valency $$\ge 3$$ would be enough). (d) is ruled out because discrete groups have no metrically proper focal action at all. So (e) holds.

The case of actions on trees is a useful motivating baby case illustrating the above "classification"; all cases can actually occur.

• Thank you! And if I assume that stabilizers are amenable, will it be the same? Mar 1 '19 at 23:40
• @MariaGerasimova you have to discard the focal case. Writing, for instance, the solvable group $\mathbf{Z}\wr\mathbf{Z}$ as an ascending HNN extension (w.r.t. an endomorphism with image of finite index), you make it act vertex-transitively on a tree with infinite valency (with abelian stabilizers). So you need to explicitly exclude the case of a fixed point at infinity.
– YCor
Mar 1 '19 at 23:56

The geometry of simplicial trees can be extended in two directions: Gromov-hyperbolic spaces and CAT(0) cube complexes. Yves' answer is based on the first point of view. About cube complexes, we have the following statement (essentially due to Caprace and Sageev):

Theorem: Let $$G$$ be a group acting on a finite-dimensional CAT(0) cube complex $$X$$. Either $$G$$ contains a non-abelian free subgroup or it contains a finite-index subgroup which decomposes as a short exact sequence $$1 \to L \to G \to \mathbb{Z}^n \to 1$$ where $$n \leq \dim(X)$$ and where $$L$$ is locally $$X$$-elliptic (ie., any finitely generated subgroup of $$L$$ has a bounded orbit in $$X$$).

As a consequence, if a Tits' alternative is known for cube-stabilisers, then it is possible to deduce a Tits' alternative in the entire group. In the particular case of trees, we have:

Corollary: Let $$G$$ be a group acting on a tree $$T$$. Either $$G$$ contains a non-abelian free subgroup or it contains a finite-index* subgroup which decomposes as a short exact sequence $$1 \to L \to G \to Z \to 1$$ where $$L$$ is locally $$T$$-elliptic and where $$Z$$ is trivial or infinite cyclic.

Of course, a direct argument can be given here.

Sketch of proof of the corollary. We distinguish two cases.

First, assume that $$G$$ has a finite orbit in $$\partial T$$. So it contains a finite-index subgroup $$H$$ which fixes a point $$\xi \in \partial T$$. Fix a basepoint $$x \in T$$. Notice that, for every $$g \in H$$, the intersection $$g [x,\xi) \cap [x,\xi)$$ contains an infinite geodesic ray (as $$g$$ fixes $$\xi$$). Consequently, $$g$$ defines a translation of length $$\tau(g)$$ on a subray of $$[x,\xi)$$. (Here $$\tau(g)$$ is positive if the translation is directed to $$\xi$$ and negative otherwise.) The key observation is that $$\tau : H \to \mathbb{Z}$$ defines a homomorphism whose kernel contains only elliptic elements (as each such element fixes pointwise a subray of $$[x,\xi)$$). Thus, we get the desired short exact sequence.

Next, assume that $$G$$ has infinite orbits in $$\partial T$$. Play ping-pong to construct a non-abelian free subgroup. $$\square$$

*As noticed by Yves Cornulier in the comments, the subgroup can be always taken with index at most two.

• You seem to assume (metric) properness at some point. Both the theorem and its corollary are false otherwise. Anyway I guess that the Caprace-Sageev theory is powerful enough to encompass non-proper actions.
– YCor
Mar 2 '19 at 9:58
• I am not assuming anything about the action. Why do you think the statements shoudn't be correct? Mar 2 '19 at 10:25
• Ah, sorry, it's confusing (at least to me) that "locally elliptic", which I usually know as an intrinsic property of the group, is used here as a property of the action, especially since the conclusion is a mixture of intrinsic properties of $G$ (the homomorphism onto an abelian group is not related to the action, as stated here) and extrinsic (being locally elliptic).
– YCor
Mar 2 '19 at 10:32
• Still there's a minor error in the corollary: it can happen that $G$ is already acting locally elliptically (horocyclic case). Side remark: in the corollary, no need to pass to a finite index subgroup, except precisely in the case $G$ preserves an axis, where one might have to pass to a subgroup of index 2.
– YCor
Mar 2 '19 at 10:34
• I replaced "elliptic" with "$X$-elliptic" to avoid confusion. I also corrected the mistake in the corollary, thank you. Mar 2 '19 at 11:03