HEIGHTS OF SUBGROUPS

Rita Gitik Eliyahu Rips

Abstract. Let H be a subgroup of a group G. We say that the height of H in G is n if for any subset {1 = g_{1}, ⋯, g_{n}} of G such that g_{i} · g_{j}^{−1} ∉ H for i ≠ j the intersection of g_{i}Hg_{i}^{−1} for i ranging from 1 to is trivial, and n is minimal with this property. We prove that any quasiconvex subgroup of a torsion-free negatively curved group has finite height. It follows that any geometrically finite subgroup of the fundamental group of a closed hyperbolic 3-manifold has finite height.

Typeset by AMS-TEX