The discriminant invariant of Cantor group actions

Dyer, Jessica; Hurder, Steven; Lukina, Olga

doi:10.1016/j.topol.2016.05.005

Cited by 26 publications

(142 citation statements)

References 26 publications

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“…However, the notion of the Ellis group does not require finite generation, and finite generation was not used in any of the proofs in . One easily checks that Theorem and all results in which we use in this paper are true for countably generated groups as well. However, some related results on strong quasi‐analyticity in may require the finite (more precisely, compact) generation assumption.…”

Section: Introductionmentioning

confidence: 98%

“…One such invariant, called the asymptotic discriminant , has been introduced by the author joint with Hurder in , as a culmination of a series of papers on actions with non‐trivial isotropy groups joint with Dyer and Hurder . The asymptotic discriminant of a minimal equicontinuous action

(X, G, Φ)

is an invariant of return equivalence of minimal Cantor systems.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem , stated below and proved in , allows us to explicitly compute the Ellis group and the isotropy subgroup of its action at a given point

x \in X

. For a given descending chain

{G_{n}}_{n ⩾ 0} = false{G_{0} \supset G_{1} \supset \dots false}, G_{0} = G,

of finite index subgroups of

G

, denote by

0true G_{\infty} = lim_{⟵} {G / G_{n + 1} \to G / G_{n}}

the inverse limit of coset spaces, with bonding maps induced by the inclusions

g G_{n + 1} \subset g G_{n}

.…”

Section: Introductionmentioning

confidence: 99%

“…There is a natural left action of

G

G_{\infty}

, given by the left multiplication

\begin{matrix} true \\ right g \cdot (g_{0} G_{0}, g_{1} G_{1}, \dots) = (g g_{0} G_{0}, g g_{1} G_{0}, \dots) . \end{matrix}

We denote such action by

(G_{\infty}, G)

. An adaptation of the result in to group actions, which can be found in [, Appendix], shows that if

(X, G, Φ)

is minimal and equicontinuous, and

x \in X

, then there exists a descending chain of finite index subgroups

{false{G_{n} false}}_{n ⩾ 0}

and a homeomorphism

ϕ : X \to G_{\infty}

, such that

ϕ (x) = e

, where

e = (e G_{n})

denotes the sequence of cosets of the identity

e \in G

, and

ϕ (g \cdot y) = g \cdot ϕ (y)

for all

y \in X

and all

g \in G

. That is,

(X, G, Φ)

and

(G_{\infty}, G)

are pointed conjugate.…”

Section: Introductionmentioning

confidence: 99%

“…The homeomorphisms

ϕ

and

\overset{ϕ}{true}

in Theorem depend on the choice of a point

x \in X

and a group chain

{false{G_{n} false}}_{n ⩾ 0}

, as discussed in more detail in . The Ellis group

\bar{normalΦ false(G false)}

depends only on the action

(X, G, Φ)

, and the isotropy group

{\bar{Φ (G)}}_{x}

depends on the choice of

x \in X

up to an isomorphism.…”

Section: Introductionmentioning

confidence: 99%

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Arboreal Cantor actions

Lukina

2018

J. London Math. Soc.

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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.

show abstract

Section: Introductionmentioning

confidence: 98%

(X, G, Φ)

is an invariant of return equivalence of minimal Cantor systems.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem , stated below and proved in , allows us to explicitly compute the Ellis group and the isotropy subgroup of its action at a given point

x \in X

. For a given descending chain

{G_{n}}_{n ⩾ 0} = false{G_{0} \supset G_{1} \supset \dots false}, G_{0} = G,

of finite index subgroups of

G

, denote by

0true G_{\infty} = lim_{⟵} {G / G_{n + 1} \to G / G_{n}}

the inverse limit of coset spaces, with bonding maps induced by the inclusions

g G_{n + 1} \subset g G_{n}

.…”

Section: Introductionmentioning

confidence: 99%

“…There is a natural left action of

G

G_{\infty}

, given by the left multiplication

\begin{matrix} true \\ right g \cdot (g_{0} G_{0}, g_{1} G_{1}, \dots) = (g g_{0} G_{0}, g g_{1} G_{0}, \dots) . \end{matrix}

We denote such action by

(G_{\infty}, G)

. An adaptation of the result in to group actions, which can be found in [, Appendix], shows that if

(X, G, Φ)

is minimal and equicontinuous, and

x \in X

, then there exists a descending chain of finite index subgroups

{false{G_{n} false}}_{n ⩾ 0}

and a homeomorphism

ϕ : X \to G_{\infty}

, such that

ϕ (x) = e

, where

e = (e G_{n})

denotes the sequence of cosets of the identity

e \in G

, and

ϕ (g \cdot y) = g \cdot ϕ (y)

for all

y \in X

and all

g \in G

. That is,

(X, G, Φ)

and

(G_{\infty}, G)

are pointed conjugate.…”

Section: Introductionmentioning

confidence: 99%

“…The homeomorphisms

ϕ

and

\overset{ϕ}{true}

in Theorem depend on the choice of a point

x \in X

and a group chain

{false{G_{n} false}}_{n ⩾ 0}

, as discussed in more detail in . The Ellis group

\bar{normalΦ false(G false)}

depends only on the action

(X, G, Φ)

, and the isotropy group

{\bar{Φ (G)}}_{x}

depends on the choice of

x \in X

up to an isomorphism.…”

Section: Introductionmentioning

confidence: 99%

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Arboreal Cantor actions

Lukina

2018

J. London Math. Soc.

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Growth and Structure of Equicontinuous Foliated Spaces

Moreira Galicia

2024

Springer Proceedings in Mathematics &Amp; Statistics

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Growth and homogeneity of matchbox manifolds

Dyer

Hurder

Lukina

2017

Indagationes Mathematicae

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Abstract. 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.

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The discriminant invariant of Cantor group actions

Cited by 26 publications

References 26 publications

Arboreal Cantor actions

Arboreal Cantor actions

Growth and Structure of Equicontinuous Foliated Spaces

Growth and homogeneity of matchbox manifolds

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