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Is the subgroup of $GL(2,\mathbb Z)$ generated by the matrices $$\left( \begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array} \right) \ \ \text{and} \ \ \left( \begin{array}{cc} 2 & 1 \\ 1 & 0 \end{array} \right)$$ free of exponential growth? More generally, how does one find all the relations between two matrices?

I am sure this is well known, so any relevant references will be appreciated.

My motivation comes from dynamical systems where these matrices specify two automorphisms of the 2-torus; I am interested in studying the orbits of their joint action.

3 added 2 characters in body

Is the subgroup of $GL(2,\mathbb Z)$ generated by the matrices $$\left( \begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array} \right) \ \ \text{and} \ \ \left( \begin{array}{cc} 2 & 1 \\ 1 & 0 \end{array} \right)$$ free?

I am sure this is well known, so any relevant references will be appreciated.

My motivation comes from dynamical systems where these matrices specify two automorphisms of the 2-torus; I am interested in studying the orbits of their joint action.

2 added 43 characters in body

Is the subgroup of $GL(2,\mathbb Z)$ generated by the matrices $$\begin{pmatrix} left( \begin{array}{cc} 1 &1 & 1 \ 1 &0\end{pmatrix}\ & 0 \end{array} \right) \ \ \text{and} \ \ \begin{pmatrix} left( \begin{array}{cc} 2 &1 & 1 \ 1 &0\end{pmatrix} & 0 \end{array} \right)$$ free?

I am sure this is well known, so any relevant references will be appreciated.

My motivation comes from dynamical systems where these matrices specify two automorphisms of the 2-torus; I am interested in studying the orbits of their joint action.

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