# Groups surjecting onto a free group

Is there something resembling a characterization of which groups can map onto a non-abelian free group? Obviously they cannot have property T, and should have nontrivial abelianization, but are there some positive results?

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Since every non-abelian free group surjects to $F_2$, an equivalent question is: What are the extensions of $F_2$? I don't see a connection to HW's large groups yet. –  Martin Brandenburg Nov 7 '11 at 10:29
Since free groups are projective, I'd say: G semidirect F(X), where F(X) is the free group on a set X, where G is an arbitrary group and where the action of F(X) on G is given by an arbitrary map X -> Aut(G). But maybe I misunderstood the question? –  Matthias Künzer Nov 7 '11 at 12:00
Igor, one positive result is that any non-abelian subgroup of a right angled Artin group (also called a graph group) surjects onto $\mathbb{F}_2$. It is a current research theme in Geometric Group Theory to show that many groups embed into right angled Artin groups (at least virtually). Thus the class of such very large'' groups is indeed quite extensive. –  Ashot Minasyan Nov 9 '11 at 14:25
@Ashot: thanks! I have seen a fair bit on RAAG, but missed this result! –  Igor Rivin Nov 9 '11 at 15:05
@Igor, this has not appeared yet, but we will hopefully post in on arXiv within the next few days. –  Ashot Minasyan Nov 9 '11 at 15:17

Such groups are often called 'very large'. A group with a very large subgroup of finite index is called 'large'. Here are some miscellaneous facts:

• Baumslag and Pride showed that every group of deficiency two (ie with a presentation with two more generators than relators) is large.

• One can deduce from Wise's residually finite version of the Rips construction that there is a 'large' version of the Rips construction; that is, for every fp group $Q$ there is a short exact sequence

$1\to K\to\Gamma\to Q\to 1$

where $K$ is 3-generated and $\Gamma$ is large.

So large (and hence very large) groups are quite common. I doubt there is any kind of characterisation, but there are some open questions that are relevant. For instance:

Question: Is there a finitely presentable group $\Gamma$ with $vb_1(\Gamma)=\infty$ which is not large?

Note that $\mathbb{Z}\wr\mathbb{Z}$ gives a non-finitely presentable counterexample. (Here $vb_1(\Gamma)$ is of course the maximum of the first Betti number over all subgroups of finite index. It is obvious that $vb_1$ of a large group is infinite.)

And another thing...

I just remembered that there is a (not implementable) algorithm to determine whether a finitely presented group is very large. The point is that the group $G=\langle x_1,\ldots,x_m\mid r_1,\ldots,r_n\rangle$ maps onto a non-abelian free group if and only if some system of equations and inequations

$[x_p,x_q] \neq 1\wedge\bigwedge_j r_j(x_1,\ldots,x_m)=1$

has a solution in $F_2$, for some $p\neq q$. Now, such systems of equations and inequations over a free group $F_n$ can be solved by Makanin's algorithm.

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Another relevant fact: Dahmani, Guirardel and Przytycki showed that a random group has property FA, and in particular is not very large. But it's conceivable that, at suitable densities, a random group is large. –  HJRW Nov 7 '11 at 10:23
I wonder what "suitable densities" means (obviously I can cook up a density, e.g., supported on free groups, but that's probably not what you have in mind...) –  Igor Rivin Nov 7 '11 at 15:58
I had in mind the density model of random groups - see the Wikipedia article on random groups. –  HJRW Nov 7 '11 at 20:58
Further to the Baumslag-Pride result, there is a result of Gromov and Stohr which says that if G=⟨X;r⟩ has only one more generator than relators but such that one of the relators is a proper power then G is large. Jack Button has done some work furthering this (he has a paper, from 2008, entitled "Large Groups of Deficiency 1"). But the proper power result is already pretty powerful - it gives you, for instance, that one-relator groups with torsion are Large. –  user6503 Nov 8 '11 at 18:10
Also, Marc Lackenby has given a characterisation of Large groups with respect to "the existence of a normal series where successive quotients are finite abelian groups with sufficiently large rank and order". The paper is "A characterisation of large finitely presented groups", J. Algebra 287 (2005) 458–473. –  user6503 Nov 8 '11 at 18:12