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We study the group      GL ⁡  ( 2 , Z ) \operatorname {GL}(2,Z)       action on a two torus      T 2 {T^2}       with Lebesgue measure. We show that any measure-preserving transformation that commutes with the group action is the action of a matrix of the form      ( m a m p ; 0 0 a m p ; m ) \left ( {\begin {array}{*{20}{c}} m &amp; 0 \\ 0 &amp; m \\ \end {array} } \right )      , where      m m       is an integer. We also show that all factors come from these commuting transformations. Finally we show that the set of self-joinings consists of the product measure and the measures sitting on the graphs      ( K u , M u ) : K = ( k a m p ; 0 0 a m p ; k ) , M = ( m a m p ; 0 0 a m p ; m ) (Ku,Mu):K = \left ( {\begin {array}{*{20}{c}} k &amp; 0 \\ 0 &amp; k \\ \end {array} } \right ),M = \left ( {\begin {array}{*{20}{c}} m &amp; 0 \\ 0 &amp; m \\ \end {array} } \right )      , and      u ∈  T 2 } u \in {T^2}\}      . We provide an example whose self-joinings consist only of the product measure and the diagonal measure.

𝐺𝐿(2,𝑍) action on a two torus