TY - JOUR
T1 - Iterated Minkowski sums, horoballs and north-south dynamics
AU - Epperlein, Jeremias
AU - Meyerovitch, Tom
N1 - Funding Information: Funding. This research was supported by a post-doctoral research fellowship from the Minerva Foundation and by the Israel Science Foundation (grant number 1052/18). Funding Information: Acknowledgements. We thank Ville Salo for valuable comments on a preliminarily version of this work, and for sharing with us examples of non-Busemann horoballs. Jeremias Epperlein thanks the Ben-Gurion University of the Negev, where this work was written, for its hospitality and stimulating atmosphere. Tom Meyerovitch thanks the University of British Columbia and the Pacific Institute of Mathematical Sciences for an excellent sabbatical, partly overlapping this work. Publisher Copyright: © 2022 European Mathematical Society.
PY - 2022/11/26
Y1 - 2022/11/26
N2 - 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 = Zd and A ⊆ Zd is a finite positively generating set containing the identity, the natural invertible extension of the map W → 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 .Γ; A/ can be recovered from the dynamics of this map, in particular, the growth type and amenability of Γ. When Γ D Zd , 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.
AB - 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 = Zd and A ⊆ Zd is a finite positively generating set containing the identity, the natural invertible extension of the map W → 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 .Γ; A/ can be recovered from the dynamics of this map, in particular, the growth type and amenability of Γ. When Γ D Zd , 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.
KW - Minkowski product
KW - Minkowski sum
KW - abstract convexity structure
KW - amenability
KW - maximum cellular automaton
UR - http://www.scopus.com/inward/record.url?scp=85151164698&partnerID=8YFLogxK
U2 - https://doi.org/10.4171/GGD/670
DO - https://doi.org/10.4171/GGD/670
M3 - Article
SN - 1661-7207
VL - 17
SP - 245
EP - 292
JO - Groups, Geometry, and Dynamics
JF - Groups, Geometry, and Dynamics
IS - 1
ER -