Non-LERFness of arithmetic hyperbolic manifold groups and mixed 3-manifold groups

Abstract

We will show that for any noncompact arithmetic hyperbolic $m$-manifold with $m>3$, and any compact arithmetic hyperbolic $m$-manifold with $m>4$ that is not a $7$-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 $3$-manifold groups. We will also show that a compact orientable irreducible $3$-manifold with empty or tori boundary supports a geometric structure if and only if its fundamental group is LERF.