Iterated Minkowski sums, horoballs and north-south dynamics

Given a finite generating set $A$ for a group $\Gamma$, we study the map $W \mapsto WA$ as a topological dynamical system -- a continuous self-map of the compact metrizable space of subsets of $\Gamma$. If the set $A$ generates $\Gamma$ 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 $\Gamma= \mathbb{Z}^d$ and $A \subseteq \mathbb{Z}^d$ a finite positively generating set containing the natural invertible extension of the map $W \mapsto W+A$ is always topologically conjugate to the unique "north-south" dynamics on the Cantor set. In contrast to this, we show that various natural "geometric" properties of the finitely generated group $(\Gamma,A)$ can be recovered from the dynamics of this map, in particular, the growth type and amenability of $\Gamma$. When $\Gamma = \mathbb{Z}^d$, 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 $\Gamma$, 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.