QUASI-CONVEX SUBGROUPS OF NEGATIVELY CURVED GROUPS

If H is a finitely generated group, then GAMMA [h1,...,h(n)], the Cayl ey graph of H with respect to a finite generating set {h1,...,h(n)}, h as as vertices the elements of H. There is an edge between the vertice s v and w of GAMMA if vh(i) = w for some i is-an-element-of {1,...,n}. Theorem. Let H be a negatively curved group. If A is an infinite quas iconvex subgroup of H (i.e. there is a real number epsilon such that e very geodesic in GAMMA between points of A is within epsilon of A) the n: (1) A has finite index in the normalizer of A in H. (2) If h is-an- element-of hAh-1 is a subset of A then hAh-1 = A. (3) If N is an infin ite normal subgroup of H and N subset-of A, then A has finite index in H. In a negatively curved group, infinite cyclic subgroups are quasic onvex. Hence (1) generalizes a theorem of Gromov's that centralizers o f elements of infinite order in a negatively curved group are virtuall y cyclic. Finitely generated free groups are negatively curved and any finitely generated subgroup of a finitely generated free group is qua siconvex. The special case of (3) where H is free and A is finitely ge nerated is well known.