Q:

decreasing chain of subgroups in the Heisenberg group

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

A:

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?

A:

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 $n_i = 2^i$.)

