Fixed points of a group action on tree

Question

Suppose a group $H$ acts on a tree $T$, and this action fixes a point. Let $T_1$ be an $H$ invariant subtree of $T$. How do I show that $H$ fixes a point in $T_1$?

Answer 1

Let $a\in T$ be the given point such that $a\cdot h=a$ for all $h\in H$. Let $x\in T_1$ such that $d(x, a)$ is minimal.

Lemma. The point $x$ is fixed by $H$.

Sketch proof. You first need to prove that if $y\in T_1$ is such that $d(x, a)=d(y, a)$ then $x=y$. To see this, you prove that there is a path $p$ from $x$ to $y$ contained in $T_1$, and hence there is a cycle which connecting the three points $x$, $a$ and $y$, a contradiction as $T$ is a tree. You need to use the minimality of $d(x, a)$ lots.

Now, as $H$ acts on $T$ by isometries, distance is preserved. Hence, for all $h\in H$ we have that $d(x\cdot h, a)=d(x, a)$. It follows from the above paragraph that $x\cdot h=x$ for all $h\in H$, as required.