If $N$ is a normal subgroup of a group $G$ such that $G/N= \mathbb{Z}$. Suppose that the classifying space of $G$ is a finite CW-complex of dimension $n$. Does it follow that the classifying space of $N$ is a finite CW-complex of dimension $n-1$ ?



Dimension of classifying space of a group
No. Baumslag-Solitar groups of type $(1, n)$, which are semidirect products $\Bbb Z \ltimes \Bbb Z[1/n]$ have finite two-dimensional classifying spaces, but $\Bbb Z[1/n]$ clearly cannot have one-dimensional classifying space.


