  Let $U$ be a horospherical subgroup of a noncompact simple Lie group $H$ and
let $A$ be a maximal split torus in the normalizer of $U$. We define the
expanding cone $A_U^+$ in $A$ with respect to $U$ and show that it can be
explicitly calculated. We prove several dynamical results for translations of
$U$-slices by elements of $A_U^+$ on finite volume homogeneous space $G/\Gamma$
where $G$ is a Lie group containing $H$. More precisely, we prove quantitative
nonescape of mass and equidistribution of a $U$-slice. If $H$ is a normal
subgroup of $G$ and the $H$ action on $G/\Gamma$ has a spectral gap, we prove
effective multiple equidistribution and pointwise equidistribution with an
error rate. In the paper we formulate the notion of the expanding cone and
prove the dynamical results above in the more general setting where $H$ is a
semisimple Lie group without compact factors. In the appendix, joint with Rene
Ruhr, we prove a multiple ergodic theorem with an error rate.
