On automorphisms of L(Z2⋊SL(2,Z)) O

We will study the automorphisms on the group II1 factor L(Z2⋊SL(2,Z)) which preserve the subalgebra L(SL(2,Z)). Our main result isOut(L(Z2⋊SL(2,Z)),L(SL(2,Z)))≃Z12⋊Z2,where Z2 acts on Z12 by the inverse operation. The proof is a modification of the recent paper due to Neshveyev and Størmer for non-commutative groups. The uniqueness of HT-Cartan subalgebras due to Popa plays a crucial role in the proof.