Proste grupe in drevesa - Pogačnik Souvent, Jakob (Matrika, 2024)
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.
Keywords: Cayley graphs; Cayleyevi grafi; drevesa; free groups; groups; grupe; proste grupe; trees