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KU Leuven（荷语鲁汶大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对virtually polycyclic群的子群，证明赫希长度不等式，明确等式成立的条件，并将其应用于扭曲共轭，建立赫希长度、重合子群与Reidemeister数有限性的关联。
AI 中文摘要
设H和K是virtually polycyclic群G的子群，我们证明赫希长度不等式：$$h(H)+h(K) \leq h(H\cap K)+h(G).$$ 我们证明当$(H,K)$双陪集的数量有限时等式成立，并且当G是幂零群时逆命题也成立。我们还将该结果应用于扭曲共轭，表明对于同态$\varphi,\psi \colon G \to H$，其中G和H都是virtually polycyclic群，赫希长度、重合子群$\mathrm{Coin}(\varphi,\psi)$与Reidemeister数$R(\varphi,\psi)$的有限性之间存在关联。
英文摘要
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.
Comments14 pages, comments welcome! v2: Added Lemma 6.1, expanded various proofs and expositions
