Let $F_2$ be the free group on two generators $x,y$ and let $R$ be the group algebra $\mathbf{Q}[F_2]$. Let $a,b,c$ be integers. Then define a right $R$-module

$M = R / (ax + by + c) R$.

I am interested in the question of whether $M$ is "detected" by finite quotients of $F_2$. That is; given $(a,b,c)$, does there exist a surjection $f$ from $F_2$ to a finite group $G$ such that $M \otimes_{\mathbf{Q}[F_2]} \mathbf{Q}[G]$ is nontrivial? There are some necessary conditions that aren't hard to check; for instance, a,b,c need to satisfy the triangle inequality with respect to any valuation on $\mathbf{Z}$. But are these conditions sufficient? For example, what if $(a,b,c) = (5,6,7)$; how would I check in this case whether $M$ is detected on some finite quotient of $F_2$?

Of course I am also interested in other elements of R, not just linear combinations of $x,y$, and $1$; but this seems like the simplest interesting case.

**Remarks:**

This question has a lot in common with this one; but in the older question, we are really asking about

*integral*group rings, where invertibility is harder to come by.The question is also related (at least in my mind) to questions about virtual positive Betti number for a finitely presented group $\Gamma$; in that case, Fox calculus gives a uniform presentation of the abelianization of the kernel of a quotient map $f: \Gamma \rightarrow G$ as a $\mathbf{Q}[G]$-module, which, if the virtual Betti number is positive, is nontrivial for some finite $G$.

One might also think of this question as having something to do with a "non-abelian Manin-Mumford conjecture." Note that if $F_2$ were replaced with $\mathbf{Z}^2$, the question would ask about the intersection between torsion points in $\mathbf{G}_m^2$ and the line $ax+by+c=0$. Manin-Mumford tells us that this intersection is finite. Is there a similar finiteness statement in this case? I guess this might say something like: is there a finite set of finite quotients $G_1, ... G_k$ such that any finite quotient of $F_2$ detecting $M$ has some $G_i$ as a quotient?

gtis justified: most likely people subscribing togtand neithergrnorrawill not be interested) $\endgroup$ – YCor Mar 25 '16 at 22:23gtand I'm interested. So for example, by taking $G$ to be the 3-element group, we get that triples of the form $a=b=c$ or $a+b+c=0$ are acceptable (those being irreducibles in $\mathbb{Q}[G]$.) In general, your condition says that, for some finite $G$ with two generators $x$ and $y$, $a+bx+cy$ is contained in a proper submodule of $\mathbb{Q}[G]$; that is, in some combination of subrepresentations of the regular representation which isn't all of them. (Just trying to reword the problem in lots of ways...) $\endgroup$ – Fedya Mar 26 '16 at 18:18