Groups where every two generator subgroup is free 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...
 A: 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.
A: There is no name, there are lots of examples. Guba gave many non-trivial examples in Guba, V. S. Conditions under which 2-generated subgroups in small cancellation groups are free.
Izv. Vyssh. Uchebn. Zaved. Mat. 1986, no. 7, 12–19 and here: Guba, V. S.
A finitely generated simple group with free 2-generated subgroups. 
Sibirsk. Mat. Zh. 27 (1986), no. 5, 50–67. There are of course infinitely generated locally free non-free groups. Every proper ascending HNN extension of a free group is an extension of such a group by a cyclic group. See also Arzhantseva, G. N. Olʹshanskiĭ, A. Yu.
Generality of the class of groups in which subgroups with a lesser number of generators are free. Mat. Zametki 59 (1996), no. 4, 489--496, 638; translation in 
Math. Notes 59 (1996), no. 3-4, 350–355. 
