  We initiate the study of the asymptotic topology of groups that can be
realized as fundamental groups of smooth complex projective varieties with
holomorphically convex universal covers (these are called here as
holomorphically convex groups). We prove the $H_1$-semistability conjecture of
Geoghegan for holomorphically convex groups. In view of a theorem of
Eyssidieux, Katzarkov, Pantev and Ramachandran \cite{ekpr}, this implies that
linear projective groups satisfy the $H_1$-semistability conjecture.
