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. ?
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
2
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.
-
$\begingroup$ In light of this, it seems plausible that, in a certain statistical sense (random group model), subgroup separability will be a property of almost all finitely presented groups (since these are almost always hyperbolic). I suspect that this is the case for Gromov's model with low density (say d < 1/5). I have already asked this in mathoverflow.net/questions/191965/… . Do you know of someone who can help with this? $\endgroup$– PabloCommented Jan 4, 2015 at 9:59
-
1$\begingroup$ @Pablo, I answered your question. It's still very unclear what happens to random groups at higher densities, which can't possibly be virtually special. $\endgroup$– HJRWCommented Jan 4, 2015 at 11:54