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We show that for k 3, given any matrix in GL.k; Z/, there is a hyperbolic fully irreducible automorphism of the free group of rank k whose induced action on Z k is the given matrix. Mathematics Subject Classification (2010). 20F65, 20E36, 20E05.doi:10.4171/cmh/257 fatcat:6fm5vfhbtncuxerft46kn5xazy