TITLE: Key Results of Combinatorial Group Theory.
QUESTION [9 upvotes]: What are the key theorems of combinatorial group theory?

By "key theorems", I mean those most commonly used in the literature.
For added context, I have copies of "Presentation of Groups," by D. L. Johnson and "Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations," by Magnus et al., and I've just started a Ph.D. in the area.

I suppose a good place to start would be

Theorem (Nielson-Shreier Theorem): Every subgroup of a free group is itself free.


Update: I've got a copy of Lyndon & Schupp's "Combinatorial Group Theory".

REPLY [5 votes]: I'm not sure where are boundaries between combinatorial group theory, low-dimensional topology, group homology and geometric group theory (and whether they exist), but here's some list. It's partly not "results", but constructions, but I think that constructions are more key-ish than theorems. (Also I have heavy bias towards homological and homotopical aspects of CGT.)

Lyndon 4-term partial resolvent of trivial module built from presentation, relation modules and $\pi_2$ of presentation complex (which is faithful $\Bbb Z[G]$-module)
(Higher) Magnus embedding and Fox free differential calculus 
Commutator calculus, Hall sets and Zassenhaus series
Weighted deficiency and infinite Golod groups
Ol'shanski theory of relative small cancellation
Various theorems about ends of groups, splittings as a graph of groups and relation with deficiency and Betti numbers
(already mentioned) Freiheitssatz and generalizations for presentations in a variety (Romanovskii, C. Gupta)
Nikolov-Segal theorem, word mappings and word width
Birkhoff's HSP theorem and its quasivariety generalization
Diamond lemma
particular Stallings theorem (if $q: G \to H$ is iso on abelianisation and epi on $H_2(-, \Bbb Z)$, then $q$ gives isomorphism on factors by LCS)
$F/[R, R]$ is torsion-free (and generalizations of this by Passi, Stohr et al)