In finitely presented groups, the question of the existence of a projective resolution $P_i$ (with each $P_i$ finitely generated) of $\mathbb{Z}G$ is equivalent to the existence of a $K(G,1)$ which has finitely many cells in each dimension. Polycyclic groups are finitely presented.

Are there polycyclic groups which are not $FP_\infty$?

As a side question, since polycylic groups are those which can be realised as subgroups of $\mathrm{GL}_n(\mathbb{Z})$, when does there exists an embedding $\imath$ of $G$ polycyclic in $\mathrm{GL}_n(\mathbb{Z})$ so that the above $K(G,1)$ may be realised as a quotient of $\mathbb{R}^n$ or some subspace of $\mathrm{GL}_n(\mathbb{Z})$ under the action of $\imath(G)$?

In finitely presented groups, the question of the existence of a projective resolution $P_i$ (with each $P_i$ finitely generated) of $\mathbb{Z}G$ is equivalent to the existence of a $K(G,1)$ which has finitely many cells in each dimension. Polycyclic groups are finitely presented.

Are there polycyclic groups which are not $FP_\infty$?

As a side question, since polycylic groups are those which can be realised as subgroups of $\mathrm{GL}_n(\mathbb{Z})$, when does there exists an embedding $\imath$ of $G$ polycyclic in $\mathrm{GL}_n(\mathbb{Z})$ so that the above $K(G,1)$ may be realised as a quotient of $\mathbb{R}^n$ or some subspace of $\mathrm{GL}_n(\mathbb{R})$ under the action of $\imath(G)$?