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According to B. Fine, G. Rosenberger, On restricted Gromov groups, Comm. Algebra 20 (1992) 2171--2181, Gromov proved the following in his long article introducing word-hyperbolic groups:

Let $x$ and $y$ be elements of a torsion-free word-hyperbolic group. Either the subgroup generated by $x$ and $y$ is cyclic, or there exists $n$ such that the subgroup generated by $x^n$ and $y^n$ is free of rank 2.

As a very special case of this, we get the following corollary

Let $x$ and $y$ be non-commuting elements in the free group on two generators. Then the subgroup generated by $x$ and $y$ contains a copy of the free group on two generators.

I am trying to cite this corollary as efficiently as possible, for background motivation in something I'm writing. Does anyone know of something slightly, erm, more accessible for the non-specialist than Gromov's original article? I don't really know any geometric group theory beyond some of the terminology and Nielsen-Schreier, but the result seems like it shouldn't be too hard to prove directly, modulo some standard results on free groups. Unfortunately, I don't really have space to sketch any proof in what I'm writing.

(So, to clarify, what I'm really hoping for is an answer saying that the result is easily deduced from material in, say, Section Z of Lyndon & Schupp or similar.)

EDIT/UPDATE: my thanks to John Stillwell and Ian Agol for pointing out what should have been blindingly obvious, namely that the result is a trivial consequence of Nielsen-Schreier, and for politely not pointing out what is just as obvious, that I should think harder before asking questions.

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# Reference request for two-generator subgroups of a free group

According to B. Fine, G. Rosenberger, On restricted Gromov groups, Comm. Algebra 20 (1992) 2171--2181, Gromov proved the following in his long article introducing word-hyperbolic groups:

Let $x$ and $y$ be elements of a torsion-free word-hyperbolic group. Either the subgroup generated by $x$ and $y$ is cyclic, or there exists $n$ such that the subgroup generated by $x^n$ and $y^n$ is free of rank 2.

As a very special case of this, we get the following corollary

Let $x$ and $y$ be non-commuting elements in the free group on two generators. Then the subgroup generated by $x$ and $y$ contains a copy of the free group on two generators.

I am trying to cite this corollary as efficiently as possible, for background motivation in something I'm writing. Does anyone know of something slightly, erm, more accessible for the non-specialist than Gromov's original article? I don't really know any geometric group theory beyond some of the terminology and Nielsen-Schreier, but the result seems like it shouldn't be too hard to prove directly, modulo some standard results on free groups. Unfortunately, I don't really have space to sketch any proof in what I'm writing.

(So, to clarify, what I'm really hoping for is an answer saying that the result is easily deduced from material in, say, Section Z of Lyndon & Schupp or similar.)