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?



Groups surjecting onto a free group
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.)




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**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. 


