Absolute profinite rigidity conjecture for lattices in ${\rm PSL}(2,\mathbb R)$ and ${\rm PSL}(2,\mathbb C)$

A lattice is a discrete subgroup of ${\rm PSL}(2,\mathbb R)$ or ${\rm PSL}(2,\mathbb C)$ having finite covolume. A group is profinitely rigid in the absolute sense if every finitely generated, residually finite group with the same finite quotients is isomorphic to it.

Absolute profinite rigidity conjecture for lattices. All lattices in

$${\rm PSL}(2,\mathbb R) \quad\text{and}\quad {\rm PSL}(2,\mathbb C)$$

are profinitely rigid in the absolute sense.

The source notes that this is widely believed for lattices in ${\rm PSL}(2,\mathbb R)$, while the consensus is less uniform in the Kleinian case. Liu's finite genus theorem gives only finitely many possibilities under an additional lattice hypothesis, and recent results constrain the remaining ambiguity.