Quantifying orbit detection: φ-order and φ-spectrum

We prove that the stable image of an endomorphism of a virtually free group is computable. For an endomorphism φ, an element $x\in G$ and a subset $K\subseteq G$, we say that the φ-order of g relative to K, $\phi {\text{-ord}}_K\left(g\right)$, is the smallest nonnegative integer k such that $g{\phi }^k\in K$. We prove that the set of orders, which we call φ-spectrum, is computable in two extreme cases: when K is a finite subset and when K is a recognizable subset. The finite case is proved for virtually free groups and the recognizable case for finitely presented groups. The case of finitely generated virtually abelian groups and some variations of the problem are also discussed.