arXiv subjects

Olga Lukina

Publications and source records attributed to Olga Lukina.

34 records · Page 2Linked to original sources

arXiv2020

Limit group invariants for non-free Cantor actions

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.

math.DS↗

arXiv2019

Orbit equivalence and classification of weak solenoids

In this work, we study minimal equicontinuous actions which are locally quasi-analytic. The first main result shows that for minimal equicontinuous actions which are locally quasi-analytic, continuous orbit equivalence of the actions implies return equivalence. This generalizes results of Cortez and Medynets, and of Li. The second main result is that if G is a finitely-generated, virtually nilpotent group, then every minimal equicontinuous action by G is locally quasi-analytic. As an application, we show that the homeomorphism type of a nil-solenoid is determined by the virtual topological full group of its monodromy action.

math.DS↗

arXiv2018

PHOTON-2017 conference proceedings

This document collects the proceedings of the PHOTON 2017 conference ("International Conference on the Structure and the Interactions of the Photon", including the 22th "International Workshop on Photon-Photon Collisions", and the "International Workshop on High Energy Photon Colliders") held at CERN (Geneva) in May 2017. The latest experimental and theoretical developments on the topics of the PHOTON conference series are covered: (i) $γ\,γ$ processes in e$^+$e$^-$, proton-proton (pp) and nucleus-nucleus (AA) collisions at current and future colliders, (ii) $γ$-hadron interactions in e$^\pm$p, pp, and AA collisions, (iii) final-state photon production (including Standard Model studies and searches beyond it) in pp and AA collisions, and (iv) high-energy $γ$-ray astrophysics. These proceedings are dedicated to the memory of Maria Krawczyk.

hep-ph↗

arXiv2018

Pro-groups and generalizations of a theorem of Bing

A matchbox manifold is a generalized lamination, which is a continuum whose path components define the leaves of a foliation of the space. A matchbox manifold is M-like if it has the shape of a fixed topological space M. When M is a closed manifold, in a previous work, the authors have shown that if $\frak M$ is a matchbox manifold which is M-like, then it is homeomorphic to a weak solenoid. In this work, we associate to a weak solenoid a pro-group, whose pro-isomorphism class is an invariant of the homeomorphism class of $\frak M$. We then show that an M-like matchbox manifold is homeomorphic to a weak solenoid whose base manifold has fundamental group which is non co-Hopfian; that is, it admits a non-trivial self-embedding of finite index. We include a collection of examples illustrating this conclusion.

math.AT↗

arXiv2018

Arboreal Cantor actions

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.

math.DS↗

arXiv2017

Classifying matchbox manifolds

Matchbox manifolds are foliated spaces with totally disconnected transversals. Two matchbox manifolds which are homeomorphic have return equivalent dynamics, so that invariants of return equivalence can be applied to distinguish non-homeomorphic matchbox manifolds. In this work we study the problem of showing the converse implication: when does return equivalence imply homeomorphism? For the class of weak solenoidal matchbox manifolds, we show that if the base manifolds satisfy a strong form of the Borel Conjecture, then return equivalence for the dynamics of their foliations implies the total spaces are homeomorphic. In particular, we show that two equicontinuous $\mT^n$--like matchbox manifolds of the same dimension are homeomorphic if and only if their corresponding restricted pseudogroups are return equivalent. At the same time, we show that these results cannot be extended to include the "\emph{adic}-surfaces", which are a class of weak solenoids fibering over a closed surface of genus 2.

math.DS↗

arXiv2017

Manifold--like matchbox manifolds

A matchbox manifold is a generalized lamination, and 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.

math.DS↗

arXiv2017

Wild solenoids

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.

math.DS↗

arXiv2016

Molino theory for matchbox manifolds

A matchbox manifold is a foliated space with totally disconnected transversals, and an equicontinuous matchbox manifold is the generalization of Riemannian foliations for smooth manifolds in this context. In this paper, we develop the Molino theory for all equicontinuous matchbox manifolds. Our work extends the Molino theory developed in the work of Álvarez López and Moreira Galicia which required the hypothesis that the holonomy actions for these spaces satisfy the strong quasi-analyticity condition. The methods of this paper are based on the authors' previous works on the structure of weak solenoids, and provide many new properties of the Molino theory for the case of totally disconnected transversals, and examples to illustrate these properties. In particular, we show that the Molino space need not be uniquely well-defined, unless the global holonomy dynamical system is tame, a notion defined in this work. We show that examples in the literature for the theory of weak solenoids provide examples for which the strong quasi-analytic condition fails. Of particular interest is a new class of examples of equicontinuous minimal Cantor actions by finitely generated groups, whose construction relies on a result of Lubotzky. These examples have non-trivial Molino sequences, and other interesting properties.

math.DS↗

arXiv2016

Growth and homogeneity of matchbox manifolds

A matchbox manifold with one-dimensional leaves which has equicontinuous holonomy dynamics must be a homogeneous space, and so must be homeomorphic to a classical Vietoris solenoid. In this work, we consider the problem, what can be said about a matchbox manifold with equicontinuous holonomy dynamics, and all of whose leaves have at most polynomial growth type? We show that such a space must have a finite covering for which the global holonomy group of its foliation is nilpotent. As a consequence, we show that if the growth type of the leaves is polynomial of degree at most 3, then there exists a finite covering which is homogeneous. If the growth type of the leaves is polynomial of degree at least 4, then there are additional obstructions to homogeneity, which arise from the structure of nilpotent groups.

math.DS↗

arXiv2016

The discriminant invariant of Cantor group actions

In this work, we investigate the dynamical and geometric properties of weak solenoids, as part of the development of a "calculus of group chains" associated to Cantor minimal actions. The study of the properties of group chains was initiated in the works of McCord 1965 and Fokkink and Oversteegen 2002, to study the problem of determining which weak solenoids are homogeneous continua. We develop an alternative condition for the homogeneity in terms of the Ellis semigroup of the action, then investigate the relationship between non-homogeneity of a weak solenoid and its discriminant invariant, which we introduce in this work. A key part of our study is the construction of new examples that illustrate various subtle properties of group chains that correspond to geometric properties of non-homogeneous weak solenoids.

math.DS↗

arXiv2014

Shape of matchbox manifolds

In this work, we develop shape expansions of minimal matchbox manifolds without holonomy, in terms of branched manifolds formed from their leaves. Our approach is based on the method of coding the holonomy groups for the foliated spaces, to define leafwise regions which are transversely stable and are adapted to the foliation dynamics. Approximations are obtained by collapsing appropriately chosen neighborhoods onto these regions along a "transverse Cantor foliation". The existence of the "transverse Cantor foliation" allows us to generalize standard techniques known for Euclidean and fibered cases to arbitrary matchbox manifolds with Riemannian leaf geometry and without holonomy. The transverse Cantor foliations used here are constructed by purely intrinsic and topological means, as we do not assume that our matchbox manifolds are embedded into a smooth foliated manifold, or a smooth manifold.

math.DS↗

arXiv2013

Suspensions of Bernoulli shifts

We show that for a given finitely generated group, its Bernoulli shift space can be equivariantly embedded as a subset of a space of pointed trees with Gromov-Hausdorff metric and natural partial action of a free group. Since the latter can be realised as a transverse space of a foliated space with leaves Riemannian manifolds, this embedding allows us to obtain a suspension of such Bernoulli shift. By a similar argument we show that the space of pointed trees is universal for compactly generated expansive pseudogroups of transformations.

math.DS↗

arXiv2012

Hierarchy of graph matchbox manifolds

We study a class of graph foliated spaces, or graph matchbox manifolds, initially constructed by Kenyon and Ghys. For graph foliated spaces we introduce a quantifier of dynamical complexity which we call its level. We develop the fusion construction, which allows us to associate to every two graph foliated spaces a third one which contains the former two in its closure. Although the underlying idea of the fusion is simple, it gives us a powerful tool to study graph foliated spaces. Using fusion, we prove that there is a hierarchy of graph foliated spaces at infinite levels. We also construct examples of graph foliated spaces with various dynamical and geometric properties.

math.DS↗

arXiv2012

Voronoi tessellations for matchbox manifolds

Matchbox manifolds ${\mathfrak M}$ are a special class of foliated spaces, which includes as special examples exceptional minimal sets of foliations, weak solenoids, suspensions of odometer and Toeplitz actions, and tiling spaces associated to aperiodic tilings with finite local complexity. Some of these classes of examples are endowed with an additional structure, that of a transverse foliation, consisting of a continuous family of Cantor sets transverse to the foliated structure. The purpose of this paper is to show that this transverse structure can be defined on all minimal matchbox manifolds. This follows from the construction of uniform stable Voronoi tessellations on a dense leaf, which is the main goal of this work. From this we define a foliated Delaunay triangulation of ${\mathfrak M}$, adapted to the dynamics of ${\mathcal F}$. The result is highly technical, but underlies the study of the basic topological structure of matchbox manifolds in general. Our methods are unique in that we give the construction of the Voronoi tessellations for a complete Riemannian manifold $L$ of arbitrary dimension, with stability estimates.

math.DS↗

arXiv2012

The Schreier continuum and ends

Blanc showed in his thesis that a compact minimal foliated space with a residual subset of 2-ended leaves can contain only 1 or 2 ended leaves. In this paper we give examples of compact minimal foliated spaces where a topologically generic leaf has 1 end, there is an uncountable set of leaves with 2 ends and a leaf with 2n ends, for a given n>1. The examples we present are weak solenoids, which allows us to represent the graph of the group action on the fibre as the inverse limit of finite coverings of a finite graph, which we call the Schreier continuum, which we use to obtain the result. While in certain cases the problem can be reduced to the study of a self-similar action of an automorphism group of a regular tree, our geometric technique is more general, as it applies to cases where the action is not self-similar.

math.DS↗