Classify finitely generated groups where twisted conjugacy growth is dominated by conjugacy growth

Determine the class of finitely generated groups G for which, for every automorphism ψ of G and every finite generating set S, the twisted conjugacy growth function Gr_{ψ,G}^S(n) (the number of ψ-twisted conjugacy classes with a representative of word norm at most n) is asymptotically dominated by the conjugacy growth function Gr_{∼G}^S(n) (the number of conjugacy classes with a representative of word norm at most n), i.e., Gr_{ψ,G}^S ≺ Gr_{∼G}^S where f ≺ g means f(n) ≤ c·g(cn) for some constant c and all n.

Background

The paper proves that, for generalised Heisenberg groups (torsion-free 2-step nilpotent groups with infinite cyclic derived subgroup), the twisted conjugacy growth is always bounded above by the ordinary conjugacy growth. Related work shows the same phenomenon for virtually abelian groups.

Motivated by these results, the authors ask for a broader classification of finitely generated groups in which the inequality Gr_{ψ,G}^S ≺ Gr_{∼G}^S holds uniformly for every automorphism ψ. They note that such a property would imply that G and any finite extension H of G share the same conjugacy growth. They also point out known non-examples and constraints from the literature, indicating that solvable and linear groups that are not virtually nilpotent satisfy a related upper bound behavior.