ON QUASI-CONVEX SUBGROUPS OF NEGATIVELY CURVED GROUPS

We say that a finitely generated group is locally quasiconvex if all i ts finitely generated subgroups are quasiconvex. Let G and H be locall y quasiconvex subgroups of a negatively curved group [GRAPHICS] and le t L be a finitely generated subgroup of [GRAPHICS] which intersects G and H in finitely generated subgroups. We prove that if G(0) is malnor mal in G and quasiconvex in [GRAPHICS] then L is quasiconvex in [GRAPH ICS] In particularly, a free product of locally quasiconvex negatively curved groups is locally quasiconvex and a free product of two negati vely curved locally quasiconvex groups amalgamated over a virtually cy clic subgroup which is malnormal in one of the factors is locally quas iconvex. We also give a new proof of the fact that locally quasiconvex groups have the finitely generated intersection property, hence the g roups mentioned above have the finitely generated intersection propert y. (C) 1997 Elsevier Science B.V.