  Let $G$ be a real centre-free semisimple Lie group without compact factors. I
prove that irreducible lattices in $G$ are rigid under two types of sublinear
distortions. The first result is that the class of lattices in groups that do
not admit $\mathbb{R}$-rank $1$ factors is $\textit{SBE complete}$: if
$\Lambda$ is an abstract finitely generated group that is Sublinearly
BiLipschitz Equivalent (SBE) to a lattice $\Gamma\leq G$, then $\Lambda$ can be
homomorphically mapped into $G$ with finite kernel and image a lattice in $G$.
For such $G$ this generalizes the well known quasi-isometric completeness of
lattices. The second result concerns sublinear distortions within $G$ itself,
and holds without any restriction on the rank of the factors: if $\Lambda\leq
G$ is a discrete subgroup that $\textit{sublinearly covers}$ a lattice
$\Gamma\leq G$, then $\Lambda$ is itself a lattice.
