Let $G$ by a hyperbolic group, and let $H \lhd G$ be a normal quasiconvex subgroup. Is it possible that $|H| = [G : H] = \infty$ ?
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7
Greenberg's theorem for hyperbolic groups, proved by Kapovich and Short, asserts that such an $H$ is finite.
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6$\begingroup$ The proof can be included: if $H$ is quasi-convex of infinite index, then $\partial H$ is a proper closed subset of $\partial G$, and $H$ normal implies that it's $G$-invariant. Since $G$ (which can be supposed non-elementary) is minimal on $\partial G$, this implies $\partial H=\emptyset$, hence $H$ is finite. $\endgroup$– YCorCommented Jan 7, 2015 at 14:14
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$\begingroup$ But isn't the proof the same in that case? $\endgroup$ Commented Jan 7, 2015 at 16:13
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$\begingroup$ Of course. But he did it before hyperbolic groups had been invented, and probably phrased his proof differently. I haven't looked at his paper. $\endgroup$– HJRWCommented Jan 7, 2015 at 16:20