NonLERFness 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 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.

31 pages, comments are welcomed!

References in corpus (3)

• Tameness of hyperbolic 3-manifolds
• Systoles of hyperbolic 4-manifolds
• Chern--Simons theory, surface separability, and volumes of 3-manifolds

Cited by in corpus (2)

• A characterization on separable subgroups of 3-manifold groups
• Alternating and Symmetric Separability of Free Products