  In this paper, we present novel and known non-relativistic and
ultra-relativistic spin-3 algebras, by considering the Lie algebra expansion
method. We start by applying the expansion procedure using different semigroups
to the spin-3 extension of the AdS algebra, leading to spin-3 extensions of
known non-relativistic and ultra-relativistic algebras. We then generalize the
procedure considering an infinite-dimensional semigroup, which allows to obtain
a spin-3 extension of two new infinite families of the Newton-Hooke type and
AdS-Carroll type. We also present the construction of the gravity theories
based on the aforementioned algebras. In particular, the expansion method based
on semigroups also allows to derive the (non-degenerate) invariant bilinear
forms, ensuring the proper construction of the Chern-Simons gravity actions.
Interestingly, in the vanishing cosmological constant limit we recover the
spin-3 extensions of the infinite-dimensional Galilean and infinite-dimensional
Carroll gravity theories.
