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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) ≤ h(H∩ 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: G → H between virtually polycyclic groups, the equivalent formulation in terms of their coincidence subgroup is h(Coin(φ,ψ)) ≥ 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.
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BibTeX
@misc{tertooy2026hirsch,
title = {{A Hirsch length inequality}},
author = {Sam Tertooy},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.16850},
eprint = {2608.16850},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.16850},
}
RIS
TY - UNPB
TI - A Hirsch length inequality
AU - Sam Tertooy
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2608.16850
UR - https://arxiv.org/abs/2608.16850
AB - Let H and K be subgroups of a virtually polycyclic group G. We prove the Hirsch length inequality h(H)+h(K) ≤ h(H∩ 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: G → H between virtually polycyclic groups, the equivalent formulation in terms of their coincidence subgroup is h(Coin(φ,ψ)) ≥ 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.
ER -