NonLERFness of arithmetic hyperbolic manifold groups and mixed 3-manifold groups

TL;DR

This paper demonstrates that most arithmetic hyperbolic manifold groups are not LERF, except for certain 3-manifolds, by analyzing abelian amalgamations and geometric structures.

Contribution

It establishes non-LERFness for broad classes of arithmetic hyperbolic manifolds and characterizes LERFness in 3-manifold groups based on geometric structures.

Findings

Noncompact arithmetic hyperbolic manifolds with m>3 have non-LERF fundamental groups.

Certain compact arithmetic hyperbolic manifolds with m>4 are not LERF, except in specific octonionic cases.

3-manifolds supporting geometric structures have LERF fundamental 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.