Deep pockets in lattices and other groups

TL;DR

This paper investigates the existence of deep pockets in groups, proving their nonexistence in many cases and providing examples of groups where deep pockets do exist, thus advancing understanding of geometric group properties.

Contribution

It extends Bogopol'skii's result by identifying classes of groups without deep pockets and presents examples of important groups that do have deep pockets.

Findings

Many groups lack deep pockets

Nil and Sol groups have deep pockets

Extension of previous nonexistence results

Abstract

We show the nonexistence of deep pockets in a large class of groups, extending a result of Bogopol'skii. We then give examples of important groups (such as Nil and Sol) which have deep pockets.