Iterated Minkowski sums, horoballs and north-south dynamics

Abstract

Given a finite generating set A for a group Γ, we study the map W → W A as a topological dynamical system -a continuous self-map of the compact metrizable space of subsets of Γ. If the set A generates Γ as a semigroup and contains the identity, there are precisely two fixed points, one of which is attracting. This supports the initial impression that the dynamics of this map is rather trivial. Indeed, at least when Γ = Z d and A ⊆ Z d a finite positively generating set containing the natural invertible extension of the map W → W + A is always topologically conjugate to the unique "northsouth" dynamics on the Cantor set. In contrast to this, we show that various natural "geometric" properties of the finitely generated group (Γ, A) can be recovered from the dynamics of this map, in particular, the growth type and amenability of Γ. When Γ = Z d , we show that the volume of the convex hull of the generating set A is also an invariant of topological conjugacy. Our study introduces, utilizes and develops a certain convexity structure on subsets of the group Γ, related to a new concept which we call the sheltered hull of a set. We also relate this study to the structure of horoballs in finitely generated groups, focusing on the abelian case.

Examples of Non-Busemann Horoballs

“…The notion of horoball is due to Gromov [8], see [7]. Our main motivation for studying horoballs is that the sets ∅ and G are also limits of suitably positioned balls of increasing radii, so the horoballs and the sets ∅ and G taken together form a G-subshift under translations, we call this the horoball subshift.…”

“…These objects arise in some symbolic dynamical applications: In [12] the author and Meyerovitch used them to study subshifts consisting of periodic points. In the recent preprint [7], Epperlein and Meyerovitch study the cellular automaton f : {0, 1} G → {0, 1} G defined by f (x) g = max s∈S (x gs ), observing that certain properties of G (such as the growth rate and amenability) can be deduced from the dynamical properties of f . The connection to horoballs is that the limit set of this cellular automaton consists of precisely the unions of horoballs, equivalently unions of Busemann horoballs.…”

“…It was asked in [7] whether on discrete groups all horoballs are Busemann. We wrote this paper somewhat in isolation in response to this question, and while it seems some of our results are not directly in the literature, most things are at least a near miss to being known -in particular, our additional discussion of ball distortion in the Heisenberg group, the most technically challenging part, can be directly deduced in the literature.…”

“…We wrote this paper somewhat in isolation in response to this question, and while it seems some of our results are not directly in the literature, most things are at least a near miss to being known -in particular, our additional discussion of ball distortion in the Heisenberg group, the most technically challenging part, can be directly deduced in the literature. As our main goal was to resolve the question of [7], we have decided to simply point out these connections rather than rewrite a shorter (possibly empty) paper with only the new observations. First, the question of [7] can be answered by appealing to the literature, as follows: A necessary condition on a horoball being Busemann is that is is connected (since Busemann horoballs are increasing unions of balls), and by compactness (Lemma 2) horoballs being connected is equivalent to distortion of intrinsic metrics of balls being bounded.…”

“…As our main goal was to resolve the question of [7], we have decided to simply point out these connections rather than rewrite a shorter (possibly empty) paper with only the new observations. First, the question of [7] can be answered by appealing to the literature, as follows: A necessary condition on a horoball being Busemann is that is is connected (since Busemann horoballs are increasing unions of balls), and by compactness (Lemma 2) horoballs being connected is equivalent to distortion of intrinsic metrics of balls being bounded. This property has been introduced previously by Cannon [3] under the name almost convexity.…”