Rationality of twisted conjugacy growth series characterizes virtually abelian finitely presented groups

Establish whether a finitely presented group G is virtually abelian if and only if, for every automorphism ψ of G and every finite generating set S, the twisted conjugacy growth series ∑_{n≥0} Gr_{ψ,G}^S(n) x^n is rational.

Background

The authors consider the twisted conjugacy growth series as an analogue of the classical conjugacy growth series. Ciobanu, Evetts and Ho conjectured that the ordinary conjugacy growth series is rational precisely for virtually abelian finitely presented groups.

As a closely related statement, the authors formulate the twisted analogue. Their main results for generalised Heisenberg groups provide evidence in favor of this conjecture, including the existence of an automorphism with transcendental twisted conjugacy growth series, suggesting that non-virtually abelian groups typically fail rationality.