Year: 2026
Type: preprint
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. For homomorphisms $φ,ψ\colon G \to H$ between virtually polycyclic groups, the equivalent formulation in terms of their coincidence subgroup is $$h(\mathrm{Coin}(φ,ψ)) \geq h(G)-h(H).$$ In this formulation, equality holds when the Reidemeister number $R(φ,ψ)$ is finite, and the converse holds when $H$ is nilpotent.
Source: arXiv (Cornell University)
Author: Tertooy, Sam
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203663968