  Perepechko and Zaidenberg conjectured that the neutral component of the
automorphism group of a rigid affine variety is a torus. We prove this
conjecture for toric varieties and varieties with a torus action of complexity
one. We also obtain a criterion for an $m$-suspension over a rigid variety to
be rigid (for every rigid variety and every regular function). Additionally, we
study the automorphism group of $m$-suspensions satisfying this criterion.
