Why does one consider the boundary of trees?

Question

this is a soft question.

Why do we /did one start to consider the boundary of trees? Was there a predominant problem with studying the trees themselves?

Answer 1

Boundaries "at infinity" of spaces and groups had already been in the air for a while, perhaps going back to the boundary of the hyperbolic plane. Then in the 1940's we had the Freudenthal-Hopf Theorem in group theory, in which the concept of the ends of a (locally compact, locally connected) topological space and the ends of a finitely generated group were used to formulate and prove:

If $G$ is a finitely generated group then the number of ends of $G$ is $0$, $1$, $2$ or $\infty$ ($\infty$ in this case meaning the cardinality of the reals).

For the special case of a free group $F_n$ with Cayley tree $T$, when we write out the conclusion of the Freudenthal-Hopf theorem we get the three cases:

• The case $n=0$, where $T$ (and $F_0=\{\text{Id}\}$) has no ends, i.e. $T$ has finite diameter, and where $T$ (and $F_0$) have empty boundary.
• The case $n=1$ where $T$ and $F_1=\mathbb Z$ have two ends, i.e. the boundary is two points;
• The case $n \ge 2$ where $T$ and $F_n$ have infinitely many ends. In particular, the boundary is a Cantor set.

From those earlier topics, eventually we had Stallings theorem about ends of groups: if a finitely generated group $G$ has infinitely many ends, then there exists a tree $T$ and a simplicial action of $G$ on $T$ such that $T$ has no $G$-invariant proper subtree, $T$ has infinitely many ends, and the action has a fundamental domain consisting of a finite subtree. This might be the real start of serious applications of the space of ends of a tree, even though the idea had already been around for a while.