Sublinear Rigidity of Lattices in Semisimple Lie Groups

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 $\mathit{\text{SBE}}\mathit{\text{complete}}$: if $\mathrm{\Lambda }$ is an abstract finitely generated group that is Sublinearly BiLipschitz Equivalent (SBE) to a lattice $\mathrm{\Gamma }\le G$, then $\mathrm{\Lambda }$ 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 $\mathrm{\Lambda }\le G$ is a discrete subgroup that $\mathit{\text{sublinearly}}\mathit{\text{covers}}$ a lattice $\mathrm{\Gamma }\le G$, then $\mathrm{\Lambda }$ is itself a lattice.