A Hirsch length inequality · arXivDeskAbstract
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
-double cosets is finite, and that the converse holds when
is nilpotent. We also apply this to twisted conjugacy, showing that for homomorphisms
φ,ψ:G→H with
and
virtually polycyclic, there is a connection between the Hirsch lengths of
,
, and the coincidence subgroup
Coin(φ,ψ) , and the finiteness of the Reidemeister number
.
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