ON QUASICONVEX SUBGROUPS

OF NEGATIVELY CURVED GROUPS

Rita Gitik

Abstract. We say that a group is locally quasiconvex if for some finite presentation all its finitely generated subgroups are quasiconvex. Let G and H be locally quasiconvex subgroups of a negatively curved group of GH (the amalgamated prodcut of G and H with G_{0} = H_{0}) and let L be a finitely generated subgroup of GH which has finitely generated intersection with any conjugate of G or of H. We prove that if G_{0} is malnormal in G and quasiconvex in GH then L is quasiconvex in A.

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 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.

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