The proper finite-index subgroup conjecture for $PD(3)$ groups

A $PD(3)$ group is a three-dimensional Poincaré duality group. A subgroup is proper if it is strictly smaller than the ambient group, and it has finite index if the quotient by its cosets is finite.

The proper finite-index subgroup conjecture. Every $PD(3)$ group contains a proper subgroup of finite index.

For 3-manifold groups, residual finiteness guarantees proper finite-index subgroups. The conjecture would imply that $PD(3)$ groups have infinite profinite completions and therefore cannot be virtually properly locally cyclic; its resolution is not given in the supplied context.