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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asked Nov 7, 2011 at 10:02
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$\begingroup$ 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. $\endgroup$– Martin BrandenburgNov 7, 2011 at 10:29
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4$\begingroup$ 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? $\endgroup$– Matthias KünzerNov 7, 2011 at 12:00
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$\begingroup$ 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. $\endgroup$– Ashot MinasyanNov 9, 2011 at 14:25
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$\begingroup$ @Ashot: thanks! I have seen a fair bit on RAAG, but missed this result! $\endgroup$– Igor RivinNov 9, 2011 at 15:05
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$\begingroup$ @Igor, this has not appeared yet, but we will hopefully post in on arXiv within the next few days. $\endgroup$– Ashot MinasyanNov 9, 2011 at 15:17
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1 Answer
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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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$\begingroup$ 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. $\endgroup$– HJRWNov 7, 2011 at 10:23
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$\begingroup$ 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...) $\endgroup$ Nov 7, 2011 at 15:58
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$\begingroup$ I had in mind the density model of random groups - see the Wikipedia article on random groups. $\endgroup$– HJRWNov 7, 2011 at 20:58
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2$\begingroup$ 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. $\endgroup$– ADLNov 8, 2011 at 18:10
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3$\begingroup$ 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. $\endgroup$– ADLNov 8, 2011 at 18:12