A Hirsch length inequality

Abstract

Let $H$ and $K$ be subgroups of a virtually polycyclic group $G$. We prove the Hirsch length inequality $$h(H)+h(K)\le h(H\cap K)+h(G).$$ We show that equality holds when the number of $(H,K)$-double cosets is finite, and that the converse holds when $G$ is nilpotent. For homomorphisms $\phi ,\psi :G\to H$ between virtually polycyclic groups, the equivalent formulation in terms of their coincidence subgroup is $$h(\mathrm{Coin}(\phi ,\psi ))\ge h(G)-h(H).$$ In this formulation, equality holds when the Reidemeister number $R(\phi ,\psi )$ is finite, and the converse holds when $H$ is nilpotent.

arXiv comment14 pages, comments welcome! v2: Added Lemma 6.1, expanded various proofs and expositions