# Lower bound for order of matrix modulo $n$

For a positive integer $n$ and a square matrix $A$ with integral entries, let $\text{ord}(A, n)$ be the smallest positive integer $k$ such that $A^k \equiv \mathbf{1} \bmod n$, if such integer $k$ exists, otherwise put $\text{ord}(A, n) := +\infty$. So $\text{ord}(A, n)$ is the order of $A$ modulo $n$.

We have the following result [1, Theorem 17].

Theorem. If $\delta > 0$ and $A$ is $2 \times 2$ matrix with determinant equal to $1$ and trace not less than $2$, then $$\text{ord}(A,n) \gg n^{1/2} \exp((\log n)^\delta),$$ for all positive integers $n$ but a set of asymptotic density $0$.

My question is: Are there other results of this kind for square matrices with arbitrary sizes and determinants? Are there stronger results?

Thanks.

 P. Kurlberg and Z. Rudnick, On Quantum Ergodicity for Linear Maps of the Torus, Commun. Math. Phys. (2001) 222, 201-227.

• The case of the $1 \times 1$ matrix $A = (2)$ already seems difficult... – js21 May 11 '17 at 15:39
• The matrix $A$ has to be hyperbolic and there exists $\delta > 0$ such that ... – jjcale May 11 '17 at 18:09