A Hirsch length inequality

Abstract

Let $H$ and $K$ be subgroups of a virtually polycyclic group $G$. We prove the Hirsch length inequality $$h(H)+h(K) \leq h(H\cap K)+h(G).$$ We show that equality holds when the number of $(H,K)$-double cosets is finite, and that the converse holds when $G$ is nilpotent. We also apply this to twisted conjugacy, showing that for homomorphisms $\varphi,\psi \colon G \to H$ with $G$ and $H$ virtually polycyclic, there is a connection between the Hirsch lengths of $G$, $H$, and the coincidence subgroup $\mathrm{Coin}(\varphi,\psi)$, and the finiteness of the Reidemeister number $R(\varphi,\psi)$.