Equicontinuity of minimal sets for amenable
                    group actions on dendrites

Shi, Enhui; Ye, Xiangdong

doi:10.1090/conm/744/14983

Cited by 5 publications

(5 citation statements)

References 17 publications

(20 reference statements)

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“…In Section 6 we show that for an amenable group G every infinite minimal set in a dendrite system (G, X) is almost automorphic. This result was strengthened recently by Shi and Ye [40], who have shown that it is actually equicontinuous. In the final section we comment on the special case where the acting group is the group of integers.…”

Section: Introductionmentioning

confidence: 71%

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Group actions on treelike compact spaces

Glasner

Megrelishvili

2019

Sci. China Math.

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We show that group actions on many treelike compact spaces are not too complicated dynamically. We first observe that an old argument of Seidler [37], implies that every action of a topological group G on a regular continuum is null and therefore also tame. As every local dendron is regular, one concludes that every action of G on a local dendron is null. We then use a more direct method to show that every continuous group action of G on a dendron is Rosenthal representable, hence also tame. Similar results are obtained for median pretrees. As a related result we show that Helly's selection principle can be extended to bounded monotone sequences defined on median pretrees (e.g., dendrons or linearly ordered sets). Finally, we point out some applications of these results to continuous group actions on dendrites.

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Section: Introductionmentioning

confidence: 71%

“…On 4 July, 2018, Shi and Ye provided a negative answer [40]: Theorem 6.6. Let G be an amenable group acting on a dendrite X.…”

Section: Tame Actions Of Amenable Groups On Dendritesmentioning

confidence: 99%

Group actions on treelike compact spaces

Glasner

Megrelishvili

2019

Sci. China Math.

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“…However, the exact analogy to the rigidity results by Witte-Morris and Deroin-Hurtado mentioned above do not hold anymore for higher rank group actions on dendrites; in fact, it is easy to see that every countable infinite group can act faithfully on a starlike dendrite of infinite order; even if the dendrite X we considered is assumed to be of finite order, there can still exist a faithful higher rank lattice action on it (see Section 6 for examples). One may consult [1,5,9,15] for some related discussions around dendrite homeomorphism groups.…”

Section: Introductionmentioning

confidence: 99%

Rigidity for higher rank lattice actions on dendrites

Shi¹,

Xu²

2022

Preprint

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We study the rigidity in the sense of Zimmer for higher rank lattice actions on dendrites and show that: (1) if Γ is a higher rank lattice and X is a nondegenerate dendrite with no infinite order points, then any action of Γ on X cannot be almost free; (2) if Γ is further a finite index subgroup of SL n (Z) with n ≥ 3, then every action of Γ on X has a nontrivial almost finite subsystem. During the proof, we get a new characterization of the left-orderability of a finitely generated group through its actions on dendrites.

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“…Some people studied the Ghys-Margulis' alternative for dendrite homeomorphism groups ( [18,15,8]). One may consult [1,12,20] for the discussions around the structures of minimal sets for group actions on dendrites. The algebraic structures of dendrite homeomorphism groups were investigated in [7] Duchesne-Monod proved the existence of finite orbits for higher rank lattice actions on dendrites ( [8]).…”

Section: Introductionmentioning

confidence: 99%

The structures of higher rank lattice actions on dendrites

Shi¹,

Xu²

2022

Preprint

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Let Γ be a higher rank lattice acting on a nondegenerate dendrite X with no infinite order points. We show that there exists a nondegenerate subdendrite Y which is Γ-invariant and satisfies the following items:(1) There is an inverse system of finite actions {(Y i , Γ) : i = 1, 2, 3, • • • } with monotone bonding maps φ i : Y i+1 → Y i and with each Y i being a dendrite, such that (Y, Γ|Y ) is topologically conjugate to the inverse limit (lim(2) The first point map r : X → Y is a factor map from (X, Γ) to (Y, Γ|Y ); if x ∈ X \ Y , then r(x) is an end point of Y with infinite orbit; for each y ∈ Y , r −1 (y) is contractible, that is there is a sequence g i ∈ Γ with diam(g i r −1 (y)) → 0.

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Equicontinuity of minimal sets for amenable group actions on dendrites

Abstract: In this note, we show that if G is an amenable group acting on a dendrite X, then the restriction of G to any minimal set K is equicontinuous, and K is either finite or homeomorphic to the Cantor set.

Cited by 5 publications

References 17 publications

Group actions on treelike compact spaces

Group actions on treelike compact spaces

Rigidity for higher rank lattice actions on dendrites

The structures of higher rank lattice actions on dendrites

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