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It is well known that a non-abelian free group is residually a finite simple group. Katz and Magnus proved, in fact, that non-abelian free groups are residually alternating and residually $PSL_{2}$. S. J. Pride has some nice results along these lines as well. The best result that I know of is the theorem of Weigel that can be formulated as follows. If $\mathfrak{X}$ is a group-theoretic class containing an infinite set of pairwise non-isomorphic finite non-abelian simple groups, then every non-abelian free group is residually an $\mathfrak{X}$-group.


My question is this:

Is a non-abelian free group fully residually a finite non-abelian simple group?

It seems likely that the answer to such an obvious question is known, but I have not been able to find it in the literature.

I should probably add that I suspect we can probably replace "finite non-abelian simple" with "alternating", but I haven't yet given any thought to the other infinite series. I'd like to learn whether anything is known before spending more time on this.

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4 Answers

up vote 6 down vote accepted

Yes! See my recent preprint Alternating quotients of free groups.

I expect that what you want is well known, but I too couldn't find it in the literature. In fact, I prove the much stronger result that free groups are something like 'locally extended fully residually alternating'. Specifically:

Let $F$ be any free group of rank at least two, let $H$ be a finitely generated subgroup of infinite index in $F$ and let $\{g_1,\ldots,g_n\}$ be a finite subset of $F\smallsetminus H$. Then there is a surjection $f$ from $F$ to a finite alternating group such that $f(g_i)$ is not in $f(H)$ for any $i$.

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Ooh, very nice! It had not occurred to me to think about the stronger result you proved. Thanks! –  James May 7 '10 at 22:53
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Indeed Thomas Weigel was the first to prove the full result. However, there is a probablistic proof due to Dixon, Pyber, Seress, and Shalev, see http://www.ams.org/mathscinet/search/publdoc.html?amp=&loc=refcit&refcit=1237075%201157915%201194786&vfpref=html&r=5&mx-pid=1971144. If you are interested in pro-p groups you might like to look at my paper http://www.ams.org/mathscinet/search/publdoc.html?amp=&loc=refcit&refcit=1971144&vfpref=html&r=11&mx-pid=1844555. Now, it has been a while since I thought about it, but I would guess that the probablistic argument will actually show also the fully residually case.

Thinking about it a bit more, I cannot see immediately how to prove the fully residually case for pro-p groups. However, if I remembr correctly the probablistc proof in the discrete case, then the fully residually case should still be fine. The issue is: given $\mathfrak{X}$ an infinite family of non isomorphic non-abelian finite simple groups, fix an identity (i.e. an element of the free group), what is the probabilty that two random elments of a group in $\mathfrak{X}$ do not satisfy the identity? If it tends to 1 with the size of the group, then the fully residually case should be true.

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Okay I have found an online version of the paper by Dixon, Pyber, Seress, and Shalev at http://mathstat.carleton.ca/~jdixon/Residual.pdf so I was able to check things. We need the following:

Theorem 3: Let $S$ be a finite simple group and let $w$ be a non-trivial element of the free group $F_2$ on $X,Y$. Then the probability that two randomly chosen elements $x$ and $y$ of $S$ satisfy both that $x$ and $y$ generate $S$ and $w(x,y) \neq 1$ tends to $1$ as $|S|$ tends $\infty$.

Now, clearly replacing $w$ by any finite set of words, the theorem is still true. Therefore, the fully residually case is true.

Now, the pro-$p$ case is more difficult because while the random generation has probability $1$ the not satisfying an identity has only positive probability. But it may be that the proof itself still works for finite number of words.

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Long and Reid's paper

Simple quotients of hyperbolic 3-manifold groups, PAMS, Volume 126, Number 3, 877–880

contains a proof of the theorem that free groups are residually PSL_2, and I think their proof shows that they are fully residually PSL_2.

At any rate, it's a nice paper.

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Their proof does indeed show that free groups are fully residually PSL_2. I should have remembered that above. –  HJRW May 7 '10 at 21:38
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