This is a follow-up to this question.
Let $H$ be a subgroup of $\mathbb{Z}^2$ with finite index $m$. Let $\phi$ be an automorphism on $\mathbb{Z}^2$. Then $\phi$ corresponds to a matrix $M$ in $ \operatorname{GL}(2, \mathbb{Z})$. Suppose that all eigenvalues of $M$ are $1$.
Question: What is the bound for the smallest $n \in \mathbb{N} \setminus \{0\}$, such that $\phi ^n(H) = H$? Is it possible to get a sublinear bound?
Thoughts: From the comment in the previous question, we know that we cannot find a sublinear bound for general $M$. I am trying to find out under what conditions we would get a sublinear bound. When $M$ is the identity we have a constant bound, I was wondering if a sublinear bound would hold for other matrices.
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