By Heisenberg group I mean the group with presentation $H$ generated by $x$ and $y$ such that $x$ and $y$ commute with $xyx^{1}y^{1}$. Is there an infinite chain of subgroups $H > H_1 > H_2 > \dots$ such that the index $[H_i: H_{i+1}]< n$ for some $n\ ?$ Thanks
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Assuming this is the discrete Heisenberg group $H=H_3({\mathbb Z})$, as in my comment above, then here is another way of looking at Mariano's answer (I think). Take any sequence of positive integers $n_1 < n_2 < \dots $ where $n_i \vert n_{i+1}$ for all $i$, and put $$ H_i = H_3(n_i{\mathbb Z}) $$ (Mariano's answer corresponds to taking $n_i = 2^i$.) 


If this is in fact the discrete integer Heisenberg group, then can't you just pass to the quotient $H / \langle [x,y] \rangle\cong \mathbb{Z}\times\mathbb{Z}$ and do it there? 

