TITLE: Groups where every two generator subgroup is free
QUESTION [6 upvotes]: Is there a name for the class of groups in the title, and any sort of characterization? Free groups and surface groups are in the class, but presumably there are many more...

REPLY [3 votes]: There is a widely used term binary finite (бинарно конечная, in Russian) group, see papers of Shunkov, Chernikov and their school. This means a group where all 2-generated subgruops are finite.
For instance, the Kourovka Notebook contains the following question (still open, as far as I know).
Question 4.74(b) (V.P.Shunkov, 1973). Does there exist a simple infinite binary finite 2-group?
Also, I stumbled across binary solvable and binary nilpotent groups... As you can guess, I am hinting that
you may call your groups 
binary free
if you like this terminology.