标题： 量化Brinkmann问题：相对$\varphi$-阶和$\varphi$-谱 显示英文标题

标题： Quantifying Brinkmann's problem: relative $\varphi$-order and $\varphi$-spectrum

摘要： 我们证明了虚拟自由群的自同态的稳定像是可计算的。 对于自同态$\varphi$，元素$x\in G$和子集$K\subseteq G$，我们说元素$g$在$K$，$\varphi\text{-ord}_K(g)$中的相对$\varphi$-阶是满足$g\varphi^k\in K$的最小非负整数$k$。 我们证明了称为$\varphi$-谱的序集在两种极端情况下是可计算的：当$K$是有限子集时，以及当$K$是可识别子集时。 有限情况对于几乎自由群已得到证明，可识别情况对于有限表示群已得到证明。 有限生成的几乎阿贝尔群的情况以及该问题的一些变体也进行了讨论。 显示英文摘要

摘要： We prove that the stable image of an endomorphism of a virtually free group is computable. For an endomorphism $\varphi$, an element $x\in G$ and a subset $K\subseteq G$, we say that the relative $\varphi$-order of $g$ in $K$, $\varphi\text{-ord}_K(g)$, is the smallest nonnegative integer $k$ such that $g\varphi^k\in K$. We prove that the set of orders, which we call $\varphi$-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.