Proste grupe in drevesa

Vsaki grupi pripada geometrijska upodobitev s Cayleyjevim grafom. V primeru prostih grup je pripadajoči graf, ob primerni izbiri generatorjev, drevo, na katerem je kanonično delovanje z levim množenjem prosto. Posledično je potrebna lastnost karakterizacije prostih grup obstoj drevesa, na katerem grupa prosto deluje. Manj očitno je, da je potrebna lastnost tudi zadostna, tj. da je vsaka grupa s prostim delovanjem na nekem drevesu prosta. Posledica te ekvivalence je dejstvo, da so podgrupe prostih grup tudi same proste.

Free groups and trees

Every group has a geometric representation in the form of a Cayley graph. For free groups with a proper choice of a generating set, the corresponding Cayley graph is a tree on which the group acts freely via a left translation action. Consequently a necessary condition for a group to be free is admitting a free action on a nonempty tree. Not immediately obvious is that this condition is also sufficient, meaning that having a free action on some tree is enough to characterise a group as free. A consequence of this equivalence is the fact that subgroups of a free group are themselves free.