Details:
Definition: A class $\mathcal{G}$ of groups satisfies the Tits alternative if for any $G$ in $\mathcal{G}$ either $G$ has a free, non-abelian subgroup or $G$ has a solvable subgroup of finite index.
The Request:
I need a reference, please, for the theorem that hyperbolic groups satisfy the Tits Alternative.
(Please see the Wikipedia article linked to above for a definition of hyperbolic groups.)
Thoughts:
My guess is that it's
Gromov, Mikhail (1987). "Hyperbolic Groups". In Gersten, S.M. (ed.). Essays in Group Theory. Mathematical Sciences Research Institute Publications, vol 8. New York, NY: Springer. pp. 75–263.
It is from the hyperbolic groups Wikipedia page. I recognise the name from his other work (I think) in geometric group theory.
However, the paper is behind a paywall.
I require the reference for my research.
Please help :)
The Tits alternative for hyperbolic groups follows from the fact that a hyperbolic group either contains a cyclic subgroup of finite index (so called "elementary hyperbolic groups") or contains a free subgroup of rank two.
Note that you can find a pdf of Gromov's essay on his website here.
The relevant result is as follows. This is a simplified version of Theorem 5.3.E of Gromov's essay (p68 of the pdf, p142 of text). A more modern citation is Proposition III.$\Gamma$.3.20 of Bridson and Haefliger, Metric spaces of non-positive curvature.
Theorem. Let $\Gamma$ be a hyperbolic group, and let $g, h\in\Gamma$ be elements of infinite order such that $\langle g, h\rangle$ is non-cyclic. Then there exists some $n\in\mathbb{Z}$ such that $\langle g^n, h^n\rangle$ is free of rank two.
Note that in order to apply the above theorem to the Tits alternative you need to prove that if $\Gamma$ is not virtually-cyclic then there exist two elements of infinite order $g, h\in\Gamma$ such that $\langle g, h\rangle$ is non-cyclic (this is clear if $\Gamma$ is torsion-free, while an afternoon with Gromov's essay or Bridson-Haefliger should clear up the "with torsion" case). However, it is extremely well-known that non-elementary hyperbolic groups contain non-abelian free subgroups so if you are just after something to cite then citing the above references would be sufficient.
Wikipedia has an elegant proof of the above theorem for torsion-free hyperbolic groups, using Ping-Pong and the action of $\Gamma$ on the boundary $\partial\Gamma$. This seems to be 8.2.E of Gromov's essay (p138/212), but Gromov is rather terse...