On quasiconvex subgroups of negatively curved groups

AbstractWe say that a finitely generated group is locally quasiconvex if all its finitely generated subgroups are quasiconvex. Let G and H be locally quasiconvex subgroups of a negatively curved group ∗G0=H0 and let L be a finitely generated subgroup of ∗G0=H0 which intersects G and H in finitely generated subgroups. We prove that if G0 is malnormal in G and quasiconvex in ∗G0=H0 then L is quasiconvex in In particularly, a free product of locally quasiconvex negatively curved groups is locally quasiconvex and a free product of two negatively curved locally quasiconvex groups amalgamated over a virtually cyclic subgroup which is malnormal in one of the factors is locally quasiconvex. We also give a new proof of the fact that locally quasiconvex groups have the finitely generated intersection property, hence the groups mentioned above have the finitely generated intersection property