A group Γ is said to be “finitely non-co-Hopfian,” or “renormalizable,” if there exists a self-embedding φ:Γ→Γ whose image is a proper subgroup of finite index. Such a proper self-embedding is called a “renormalization for Γ.” In this work, we associate a dynamical system to a renormalization φ of Γ. The discriminant invariant Dφ of the associated Cantor dynamical system is a profinite group which is a measure of the asymmetries of the dynamical system. If Dφ is a finite group for some renormalization, we show that Γ/Cφ is virtually nilpotent, where Cφ is the kernel of the action map. We introduce the notion of a (virtually) renormalizable Cantor action, and show that the action associated to a renormalizable group is virtually renormalizable. We study the properties of virtually renormalizable Cantor actions, and show that virtual renormalizability is an invariant of continuous orbit equivalence. Moreover, the discriminant invariant of a renormalizable Cantor action is an invariant of continuous orbit equivalence. Finally, the notion of a renormalizable Cantor action is related to the notion of a self-replicating group of automorphisms of a rooted tree.
Steven Hurder
1
;
Olga Lukina
2
;
Wouter van Limbeek
1
1
University of Illinois at Chicago, USA
2
University of Vienna, Austria
Steven Hurder; Olga Lukina; Wouter van Limbeek. Cantor dynamics of renormalizable groups. Groups, geometry, and dynamics, Tome 15 (2021) no. 4, pp. 1449-1487. doi: 10.4171/ggd/636
@article{10_4171_ggd_636,
author = {Steven Hurder and Olga Lukina and Wouter van Limbeek},
title = {Cantor dynamics of renormalizable groups},
journal = {Groups, geometry, and dynamics},
pages = {1449--1487},
year = {2021},
volume = {15},
number = {4},
doi = {10.4171/ggd/636},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/636/}
}
TY - JOUR
AU - Steven Hurder
AU - Olga Lukina
AU - Wouter van Limbeek
TI - Cantor dynamics of renormalizable groups
JO - Groups, geometry, and dynamics
PY - 2021
SP - 1449
EP - 1487
VL - 15
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.4171/ggd/636/
DO - 10.4171/ggd/636
ID - 10_4171_ggd_636
ER -
%0 Journal Article
%A Steven Hurder
%A Olga Lukina
%A Wouter van Limbeek
%T Cantor dynamics of renormalizable groups
%J Groups, geometry, and dynamics
%D 2021
%P 1449-1487
%V 15
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4171/ggd/636/
%R 10.4171/ggd/636
%F 10_4171_ggd_636
