Galois groups and Cantor actions Lukina, Olga
Transactions of the American Mathematical Society,
03/2021, Volume:
374, Issue:
3
Journal Article
Peer reviewed
Open access
In this paper, we study the actions of profinite groups on Cantor sets which arise from representations of Galois groups of certain fields of rational functions. Such representations are associated ...to polynomials, and they are called profinite iterated monodromy groups. We are interested in a topological invariant of such actions called the asymptotic discriminant. In particular, we give a complete classification by whether the asymptotic discriminant is stable or wild in the case when the polynomial generating the representation is quadratic. We also study different ways in which a wild asymptotic discriminant can arise.
Nilpotent Cantor actions Hurder, Steven; Lukina, Olga
Proceedings of the American Mathematical Society,
01/2022, Volume:
150, Issue:
1
Journal Article
Peer reviewed
Open access
A nilpotent Cantor action is a minimal equicontinuous action \Phi \colon \Gamma \times \mathfrak {X} \to \mathfrak {X} on a Cantor space \mathfrak {X}, where \Gamma contains a finitely-generated ...nilpotent subgroup \Gamma _0 \subset \Gamma of finite index. In this note, we show that these actions are distinguished among general Cantor actions: any effective action of a finitely generated group on a Cantor space, which is continuously orbit equivalent to a nilpotent Cantor action, must itself be a nilpotent Cantor action. As an application of this result, we obtain new invariants of nilpotent Cantor actions under continuous orbit equivalence.
A Cantor action is a minimal equicontinuous action of a countably generated group
$G$
on a Cantor space
$X$
. Such actions are also called generalized odometers in the literature. In this work, we ...introduce two new conjugacy invariants for Cantor actions, the stabilizer limit group and the centralizer limit group. An action is wild if the stabilizer limit group is an increasing sequence of stabilizer groups without bound and otherwise is said to be stable if this group chain is bounded. For Cantor actions by a finitely generated group
$G$
, we prove that stable actions satisfy a rigidity principle and furthermore show that the wild property is an invariant of the continuous orbit equivalence class of the action. A Cantor action is said to be dynamically wild if it is wild and the centralizer limit group is a proper subgroup of the stabilizer limit group. This property is also a conjugacy invariant and we show that a Cantor action with a non-Hausdorff element must be dynamically wild. We then give examples of wild Cantor actions with non-Hausdorff elements, using recursive methods from geometric group theory to define actions on the boundaries of trees.
Wild solenoids Hurder, Steven; Lukina, Olga
Transactions of the American Mathematical Society,
04/2019, Volume:
371, Issue:
7
Journal Article
Peer reviewed
Open access
A weak solenoid is a foliated space defined as the inverse limit of finite coverings of a closed compact manifold M. The monodromy of a weak solenoid defines an equicontinuous minimal action on a ...Cantor space X by the fundamental group G of M. The discriminant group of this action is an obstruction to this action being homogeneous. The discriminant vanishes if the group G is abelian, but there are examples of actions of nilpotent groups for which the discriminant is non-trivial. The action is said to be stable if the discriminant group remains unchanged for the induced action on sufficiently small clopen neighborhoods in X. If the discriminant group never stabilizes as the diameter of the clopen set U tends to zero, then we say that the action is unstable, and the weak solenoid which defines it is said to be wild. In this work, we show two main results in the course of our study of the properties of the discriminant group for Cantor actions. First, the tail equivalence class of the sequence of discriminant groups obtained for the restricted action on a neighborhood basis system of a point in X defines an invariant of the return equivalence class of the action, called the asymptotic discriminant, which is consequently an invariant of the homeomorphism class of the weak solenoid. Second, we construct uncountable collections of wild solenoids with pairwise distinct asymptotic discriminant invariants for a fixed base manifold M, and hence fixed finitely-presented group G, which are thus pairwise non-homeomorphic. The study in this work is the continuation of the seminal works on homeomorphisms of weak solenoids by Rogers and Tollefson in 1971 and is dedicated to the memory of Jim Rogers.
Manifold-like matchbox manifolds Clark, Alex; Hurder, Steven; Lukina, Olga
Proceedings of the American Mathematical Society,
08/2019, Volume:
147, Issue:
8
Journal Article
Peer reviewed
Open access
A matchbox manifold is a generalized lamination, which is a continuum whose arc-components define the leaves of a foliation of the space. The main result of this paper implies that a matchbox ...manifold which is manifold-like must be homeomorphic to a weak solenoid.
The paper considers the possibilities, prospects, and drawbacks of the mixed reality (MR) technology application using mixed reality smartglasses Microsoft HoloLens 2. The main challenge was to find ...and develop an approach that would allow surgeons to conduct operations using mixed reality on a large scale, reducing the preparation time required for the procedure and without having to create custom solutions for each patient. Research was conducted in three clinical cases: two median neck and one branchial cyst excisions. In each case, we applied a unique approach of hologram positioning in space based on mixed reality markers. As a result, we listed a series of positive and negative aspects related to MR surgery, along with proposed solutions for using MR in surgery on a daily basis.
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8.
Wild Cantor actions ÁLVAREZ LÓPEZ, Jesús; BARRAL LIJO, Ramon; LUKINA, Olga ...
Journal of the Mathematical Society of Japan,
04/2022, Volume:
74, Issue:
2
Journal Article
Peer reviewed
Open access
The discriminant group of a minimal equicontinuous action of a group G on a Cantor set X is the subgroup of the closure of the action in the group of homeomorphisms of X, consisting of homeomorphisms ...which fix a given point. The stabilizer and the centralizer groups associated to the action are obtained as direct limits of sequences of subgroups of the discriminant group with certain properties. Minimal equicontinuous group actions on Cantor sets admit a classification by the properties of the stabilizer and centralizer direct limit groups. In this paper, we construct new families of examples of minimal equicontinuous actions on Cantor sets, which illustrate certain aspects of this classification. These examples are constructed as actions on rooted trees. The acting groups are countable subgroups of the product or of the wreath product of groups. We discuss applications of our results to the study of attractors of dynamical systems and of minimal sets of foliations.
Arboreal Cantor actions Lukina, Olga
Journal of the London Mathematical Society,
June 2019, 2019-06-00, Volume:
99, Issue:
3
Journal Article
Peer reviewed
In this paper, we consider minimal equicontinuous actions of discrete countably generated groups on Cantor sets, obtained from the arboreal representations of absolute Galois groups of fields. In ...particular, we study the asymptotic discriminant of these actions. The asymptotic discriminant is an invariant obtained by restricting the action to a sequence of nested clopen sets, and studying the isotropies of the enveloping group actions in such restricted systems. An enveloping (Ellis) group of such an action is a profinite group. A large class of actions of profinite groups on Cantor sets is given by arboreal representations of absolute Galois groups of fields. We show how to associate to an arboreal representation an action of a discrete group, and give examples of arboreal representations with stable and wild asymptotic discriminant.
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We study the Hausdorff and the box dimensions of closed invariant subsets of the space of pointed trees, equipped with a pseudogroup action. This pseudogroup dynamical system can be regarded as a ...generalization of a shift space. We show that the Hausdorff dimension of this space is infinite, and the union of closed invariant subsets with dense orbit and non-equal Hausdorff and box dimensions is dense in this space.
We apply our results to the problem of embedding laminations into differentiable foliations of smooth manifolds. One of necessary conditions for the existence of such an embedding is that the lamination must admit a bi-Lipschitz embedding into a manifold. A suspension of the pseudogroup action on the space of pointed graphs gives an example where this condition is not satisfied, with Hausdorff dimension of the space of pointed trees being the obstruction to the existence of such a bi-Lipschitz embedding.
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