# Action of a subgroup of finite index on a tree induced by an action of a group on a tree

Let $G$ be a group wich acts on a tree $\Gamma$. Then $U$ acts on $\Gamma$ for every $U\leq G$.

Question: Why does the following hold?

If $|G:U|<\infty$. Then the minimal $U$-invariant subtree of $\Gamma$ coincides with the minimal $G$-invariant subtree of $\Gamma$. (minimal always with respect to inclusion)

Thanks for help.

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The question looks interesting but I think I'm missing some definitions, what do you mean by coincides and does the definition of a group action on a graph differ much from the definition for one on a set? Must adjacency be preserved by a fixed group element? Thanks – muzzlator Feb 4 '13 at 10:08
@user60855: There is a very well developed theory on groups acting on trees. The classic text is the book "Trees" by Serre (originally in French, if you speak it) but there is a gentle introduction by Meier called "Graphs, Groups and Trees" which is rather good. – user1729 Feb 4 '13 at 10:53

(1) some $g\in G$ acts hyperbolically. Then the minimal invariant subtree is the (convex hull of the) union of axes of hyperbolic elements and this does not change when passing to a finite index subgroup.
If $G$ is finitely generated, that's it. But otherwise there's:
(3) ("horocyclic" case) the action has no hyperbolic element, but is unbounded (so there's a unique fixed point at infinity). Then there is no minimal invariant tree for $G$ or for its finite index subgroups.