Absolute profinite rigidity and hyperbolic geometry

Abstract

We prove that there exist finitely presented, residually finite groups that are profinitely rigid in the class of all finitely presented groups but not in the class of all finitely generated groups. These groups are of the form Γ × Γ where Γ is a profinitely rigid 3-manifold group; we describe a family of such groups with the property that if P is a finitely generated, residually finite group with P ∼ = Γ × Γ then there is an embedding P ֒→ Γ × Γ that induces the profinite isomorphism; in each case there are infinitely many non-isomorphic possibilities for P .

On the profinite rigidity of triangle groups

“…More formally, if is finitely generated and residually finite, then implies (where denotes the profinite completion of a group ). Finitely generated abelian groups have this property, as do certain nilpotent groups, but it is hard to construct examples of profinitely rigid groups that do not satisfy a group law; indeed no such groups were known until our work in [6]. The most compelling question in the field is the conjecture that non‐abelian free groups of finite rank are profinitely rigid.…”

“…In [6], we proved that certain arithmetic lattices in are profinitely rigid, including the Bianchi group (where ) and the fundamental group of the Weeks manifold, which is the closed hyperbolic 3‐manifold of minimal volume. Our main purpose in the present article is to prove that certain arithmetic lattices in are also profinitely rigid in the absolute sense.…”

“…In its broad outline, our strategy for proving Theorem 1.1 is the one employed in [6] to establish the existence of Kleinian groups that are profinitely rigid. Several of the key ideas developed in [6], notably that of Galois rigidity , will play a crucial role again here.…”

“…In its broad outline, our strategy for proving Theorem 1.1 is the one employed in [6] to establish the existence of Kleinian groups that are profinitely rigid. Several of the key ideas developed in [6], notably that of Galois rigidity , will play a crucial role again here. But the end‐game by which we move from the general construction of representations to profinite rigidity for specific examples is more direct in the present setting than it was in [6].…”

“…Several of the key ideas developed in [6], notably that of Galois rigidity , will play a crucial role again here. But the end‐game by which we move from the general construction of representations to profinite rigidity for specific examples is more direct in the present setting than it was in [6].…”