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



decreasing chain of subgroups in the Heisenberg group
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 $n\_i = 2^i$.)


