Bianchi groups are conjugacy separable

Abstract

a b s t r a c tWe prove that non-uniform arithmetic lattices of SL 2 (C) and consequently the Bianchi groups are conjugacy separable. The proof is based on recent deep results of Agol, Long, Reid and Minasyan. The conjugacy separability of groups commensurable with limit groups is also established.

3-manifold groups

“…For this reason, one defines a group to be hereditarily conjugacy separable if every finite-index subgroup is conjugacy separable. We will now show that, in the 3-manifold context, one can apply a criterion of Chagas-Zalesskii [ChZ10] to prove that hereditary conjugacy separability passes to finite extensions.…”

Separability of double cosets and conjugacy classes in 3-manifold groups

“…The crucial concept is the notion of a special group, introduced by Haglund and Wise [13]. † Minasyan proved that special groups are conjugacy separable [24], while Chagas and the third author gave conditions under which conjugacy separability passes to finite extensions [10]. Taking these two results together, it follows that, if N is a hyperbolic 3-manifold and π 1 (N ) is virtually special (that is, π 1 (N ) has a special subgroup of finite index), then π 1 (N ) is conjugacy separable.…”

“…Many hyperbolic 3-manifold and orbifold groups are known to be virtually special [7,8,10,11], among them non-cocompact arithmetic and standard cocompact arithmetic lattices. Wise has announced a proof that the fundamental group of any hyperbolic 3-manifold containing an embedded geometrically finite surface is virtually special [31]; the heart of his proof is contained in [32].…”

Virtually compact special hyperbolic groups are conjugacy separable

“…In [25] the first author showed that right angled Artin groups are hereditarily conjugacy separable. This result was subsequently used to prove conjugacy separability of Bianchi groups [7], 1-relator groups with torsion [26] and fundamental groups of compact 3-manifolds [17]. In fact, in [25] it was shown that any virtually compact special group G contains a conjugacy separable subgroup of finite index.…”