  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 arithmetic hyperbolic manifold defined by octonions, its
fundamental group is not 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.

Geometric Topology;Group Theory