AbstractIf H is a finitely generated group, then Γ〈h1,…,hn〉, the Cayley graph of H with respect to a finite generating set {h1,…,hn}, has as vertices the elements of H. There is an edge between the vertices v and w of Γ if vhi = w for some i ∈ {1,…,n}.Theorem. Let H be a negatively curved group. If A is an infinite quasiconvex subgroup of H (i.e. there is a real number ε such that every geodesic in Γ between points of A is within ε of A) then:(1) A has finite index in the normalizer of A in H.(2) If h ε H and hAh-1 is a subset of A then hAh-1 = A.(3) If N is an infinite normal subgroup of H and N ⊂ A, then A has finite index in H.In a negatively curved group, infinite cyclic subgroups are quasiconvex. Hence (1) generalizes a theorem of Grom...