Automorphism groups of semi-rigid $T$-varieties of complexity one

Abstract: We prove that if $X$ is a semi-rigid normal rational affine algebraic variety with torus action of complexity one, with only constant invertible global functions and finitely generated divisor class group, then the identity component of the automorphism group of $X$ is a semi-direct product of a torus and the maximal unipotent-generated subgroup.