| The depth of a group element g is defined as the distance from g to the complement of the radius- d(1, g) closed ball about the identity. If the depth is unbounded as g ranges over all group elements, then we say the group has deep pockets. Note that these properties depend on the generating set with respect to which distance is measured. We show that neither groups with a regular language of geodesics nor euclidean groups have deep pockets. In contrast, the Heisenberg group has deep pockets with respect to every generating set. However, define the retreat depth of a group element g to be the minimal l such that g lies in an unbounded component of the complement of the radius-d(1, g) -- l closed ball about the identity. Then we show that the Heisenberg group has bounded retreat depth with respect to every generating set. Also, we show that regular lattices in Sol and the soluble Baumslag-Solitar groups have deep pockets with respect to an appropriate generating set for each. This proof involves the new notion of strong t-logarithmicity. In contrast, we construct generating sets for the lamplighter groups with respect to which they lack deep pockets. Finally, we introduce the property of having semilocally connected spheres (a weaker version of almost convexity) and show that Z2&m22;Z lacks this property, while BS(1, 2) has it. |
