2023/01/12 by Ido Grayevsky, Grayevsky, Ido
Mathematics · Physics and Astronomy · #20F38 #20F65 #20F67 #20F69 #22E15 #22E40 #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Metric Geometry (math.MG) #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2301.05086
openalex publication_date 2023/01/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ℝ-rank 1 factors is SBE complete: if Λ is an abstract finitely generated group that is Sublinearly BiLipschitz Equivalent (SBE) to a lattice Γ≤ G, then Λ 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 Λ≤ G is a discrete subgroup that sublinearly covers a lattice Γ≤ G, then Λ is itself a lattice.