Pro-p completions of groups of cohomological dimension 2

We study when an abstract finitely presented group G of cohomological dimension cd(G) = 2 has pro-p completion (G) over cap (p) of cohomological dimension cd((G) over cap (p)) <= 2. Furthermore, we prove that for a tree hyperbolic limit group G we have cd((G) over cap (p)) <= 2 and show an example of a hyperbolic limit group G that is not free and (G) over cap (p) is free pro-p. For a finitely generated residually free group G that is not a limit group, we show that (G) over cap (p) is not free pro-p.