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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
up vote 13 down vote accepted

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. – ADL 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. – ADL Nov 8 '11 at 18:12

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