The asymptotic density of dead ends in non-amenable groups
We show that, in non-amenable groups, the density of elements of depth at least $d$ goes to $0$ exponentially in $d$.
math.GR↗
arXiv subjects
Andrew D. Warshall
Publications and source records attributed to Andrew D. Warshall.
Pairs of valuations and the geometry of soluble groups
We introduce the concepts of a pair of valuations and a good generating set and show how they can be used to prove geometric properties of soluble groups.
math.GR↗
A group with deep pockets for all finite generating sets
We show that the discrete Heisenberg group has unbounded dead-end depth with respect to every finite generating set. We also show that, in contrast, it has bounded retreat depth.
math.GR↗
Deep pockets in lattices and other groups
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.
math.GR↗
Arbitrarily large finite quotients imply no uniform bound on depth
We show that any group with arbitrarily large finite quotients admits generating sets with respect to which it has arbitrarily large finite dead-end depth. This extends a joint result with Riley and partially answers a question asked there.
math.GR↗
The unbounded dead-end depth property is not a group invariant
The dead-end depth of an element g of a group with finite generating set A is the distance from g to the complement of the radius d(1,g) closed ball, in the word metric d associated to A. We exhibit a finitely presented group K with two finite generating sets A and B such that dead-end depth is unbounded on K with respect to A but is at most two with respect to B.
math.GR↗