We will show that for any noncompact arithmetic hyperbolic -manifold with , and any compact arithmetic hyperbolic -manifold with that is not a -dimensional one defined by octonions, its fundamental group is not locally extended residually finite (LERF). The main ingredient in the proof is a study on abelian amalgamations of hyperbolic -manifold groups. We will also show that a compact orientable irreducible -manifold with empty or tori boundary supports a geometric structure if and only if its fundamental group is LERF.
"Non-LERFness of arithmetic hyperbolic manifold groups and mixed -manifold groups." Duke Math. J. 168 (4) 655 - 696, 15 March 2019. https://doi.org/10.1215/00127094-2018-0048