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 \(\textit{SBE complete}\): 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.