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$H$ is called a nearly normal subgroup of a group $G$ if it is of finite index in its normal closure, $H^G:=\cup_{g\in G} gHg^{-1}$, in $G$. Clearly, every normal subgroup or subgroup of finite index of $G$ is nearly normal. I am not interested in finite subgroups either. I am looking for infinite nearly normal subgroups of infinite index in well-known groups. By a well-known group, I mean a group that some of its basic properties like amenability, growth or its geometric nature (the spaces it acts) have been studied in the literature.

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Take any infinite group with an infinite-index normal subgroup. Say, the free group on two generators with the commutator subgroup. Then take a finite index subgroup of the normal subgroup. This particular normal subgroup is the free group on countably infinite generators, and so has many finite-index subgroups. Most are not normal, but all are nearly normal.

This feels somewhat unsatisfying, though. Presumably the only way to get something satisfying is to find a nearly normal subgroup whose definition is more natural than the corresponding normal subgroup. I do not know any subgroups like that.

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@ Will: Thank you for your answer. It is stimulating. But I am looking for very concrete examples. For instance, is $SL(2,\mathbb{Z})$ nearly normal in $GL(2,\mathbb{Q})$? –  Vahid Shirbisheh May 22 '12 at 17:27
    
@Vahid: SL(2, Q) is simple modulo its center, so SL(2,Z) is not nearly normal in GL(2,Q). If you want concrete examples, take any finite index non-normal subgroup H<G_1. Then H is nearly normal in G=G_1\times G_1. For instance, take G_1=SL(2,Z) and H<G_1 be the subgroup of matrices which reduce to upper-triangular modulo some prime p. –  Misha May 22 '12 at 17:50
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