Is there a finitely generated non-elementary word hyperbolic group the profinite completion of which is known (or conjectured) to be rather restricted, that is: abelian, pro-$p$, virtually prosolvable, etc. ?



A hyperbolic group with a small profinite completion
It's a famous open question whether every word-hyperbolic group is residually finite. Kapovich--Wise showed that this is equivalent to asking whether every non-trivial word-hyperbolic group has non-trivial profinite completion. In the other direction, Agol--Groves--Manning showed that it's equivalent to asking whether every quasiconvex subgroup of every word-hyperbolic group is separable.


In particular, since non-elementary word-hyperbolic groups always contain quasiconvex non-abelian free subgroups, if an example were known with virtually prosolvable profinite completion, the above questions would all be answered in the negative.


The best conjectural candidates for non-elementary word-hyperbolic groups with restricted profinite completion (in some sense) are cocompact lattices in $Sp(n,1)$ (aka quaternionic hyperbolic lattices). It's unknown whether these have the congruence subgroup property, and if they do, then I believe it again follows that all the above questions are answered in the negative.


