Hyperbolic groups are not simple

Question

I read that a non-elementary hyperbolic group $G$ always has a normal subgroup $H\leqslant G$ such that $H$ and $G/H$ are both infinite.

(1) Any reference for a proof?

(2) Can $H$ be chosen quasi-convex?

Answer 1

For (2), no infinite, infinite index quasiconvex subgroup is normal: if $H$ is quasi-convex and you conjugate $H$ by an element $g \in G$ that is far from $H$ then $gHg^{-1} \ne H$.

Answer 2

For (1) see:

A. Yu. Olshanski, SQ-universality of hyperbolic groups, (Russian) Mat. Sb. 186 (1995), no. 8, 119–132; English translation in Sb. Math. 186 (1995), no. 8, 1199–1211.

T. Delzant, Sous-groupes distingues et quotients des groupes hyperboliques, Duke Math. J. 83 (1996), no. 3, 661–682.