# 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...

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If you have a closed hyperbolic 3-manifold $M$ with $dim H_1(M;F)\geq 3$ for a field $F$, then $\pi_1(M)$ has this property. –  Ian Agol Apr 15 '12 at 4:01
I believe that this holds for any fully residually free group. I think G. Baumslag proved it. –  Benjamin Steinberg Apr 15 '12 at 4:27
Non abelian of course! –  Benjamin Steinberg Apr 15 '12 at 4:28
@Agol: so presumably this holds for a subgroup of finite index in any closed hyp. manifold? –  Igor Rivin Apr 15 '12 at 4:45
Igor - Apparently Culler and Shalen call this property "2-free" - there are quite a few papers relating $k$-freeness and volume of hyperbolic three-manifolds. Eg google the phrase "k-free shalen" without quotes. Cheers! –  Sam Nead Apr 15 '12 at 18:38

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.

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This is really great phrase "there is no name, there are lots of examples" :-) –  Victor Apr 18 '12 at 7:54

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.

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