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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?

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.

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.

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.

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.

Is the subgroup of $GL(2,\mathbb Z)$ generated by the matrices $$ \begin{pmatrix} 1&1 \\ 1&0\end{pmatrix}\ \ \text{and} \ \ \begin{pmatrix} 2&1 \\ 1&0\end{pmatrix} $$ 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.

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.