New Techniques for Computing Geometric Index

Andrist, Kathryn B.; Garity, Dennis J.; Repovš, Dušan; Wright, David

doi:10.1007/s00009-017-1036-1

Cited by 5 publications

(17 citation statements)

References 15 publications

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“…By Theorem 1.3, the geometric index of T ′ 3 in h(T 1 ) is even. Since the geometric index of T ′ 3 in T 3 is two [1], the geometric index of T ′ 3 in h(T 1 ) cannot be 0 and so is at least two. Theorem 1.2 now implies that the geometric index of T ′ 3 in T 3 is at least 4, which is a contradiction.…”

Section: Appendixmentioning

confidence: 99%

Free groups as end homogeneity groups of $3$-manifolds

Garity,

Repovš

2022

Preprint

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For every finitely generated free group F , we construct an irreducible open 3-manifold M F whose end set is homeomorphic to a Cantor set, and with the end homogeneity group of M F isomorphic to F . The end homogeneity group is the group of all selfhomeomorphisms of the end set that extend to homeomorphisms of the entire 3-manifold. This extends an earlier result that constructs, for each finitely generated abelian group G, an irreducible open 3-manifold M G with end homogeneity group G. The method used in the proof of our main result also shows that if G is a group with a Cayley graph in R 3 such that the graph automorphisms have certain nice extension properties, then there is an irreducible open 3-manifold M G with end homogeneity group G.

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Section: Appendixmentioning

confidence: 99%

Free groups as end homogeneity groups of $3$-manifolds

Garity,

Repovš

2022

Preprint

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“…(iii) If (i) and (ii) do not hold, then infinitely many of the m j satisfy m j ≥ 2 (and m j ≥ 1 for every j). Then by grouping the arrows in (1) in such a way that each group contains at least one element ≥ 2 we may assume that m j ≥ 2 for every j. It is then very easy to see that G is not finitely generated (the argument is essentially the same as in [2, Proposition 1.6]).…”

Section: Some Algebraic Preliminariesmentioning

confidence: 99%

“…It is also clear from the construction that each γ j with j ≤ w has winding number 1 in V 0 while each γ j with j > w has winding number 0, so that γ has winding number w in V 0 . It only remains to check that the geometric index of γ in V 0 is precisely 2k + w. To see this we apply a technique of Andrist, Garity, Repovš and Wright [1]. The dotted radial lines in Figure 1.…”

Section: Toroidal Sets With Prescribed Cohomology and Self-indexmentioning

confidence: 99%

“…(b) represent two meridional disks which decompose the solid torus V 0 into two sectors C 1 and C 2 , where C 1 is shaded light gray. Sectors of this form are called chambers in [1]. For 1 ≤ j ≤ w the intersection γ j ∩ C 1 just consists of an arc that runs from one of the meridional disks to the other without turning back over.…”

Section: Toroidal Sets With Prescribed Cohomology and Self-indexmentioning

confidence: 99%

“…(1) None of the (unknotted) generalized solenoids of Example 3.4 satisfies N ∼ 1; thus, none of them is weakly tame.…”

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confidence: 99%

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The geometric index and attractors of homeomorphisms of $\mathbb{R}^3$

Barge¹,

Sánchez-Gabites²

2019

Preprint

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In this paper we focus on compacta K ⊆ R 3 which possess a neighbourhood basis that consists of nested solid tori T i . We call these sets toroidal. In [2] we defined the genus of a toroidal set as a generalization of the classical notion of genus from knot theory.Here we introduce the self-geometric index of a toroidal set K, which captures how each torus T i+1 winds inside the previous T i . We use this index in conjunction with the genus to approach the problem of whether a toroidal set can be realized as an attractor for a flow or a homeomorphism of R 3 . We obtain a complete characterization of those that can be realized as attractors for flows and exhibit uncountable families of toroidal sets that cannot be realized as attractors for homeomorphisms.

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The geometric index and attractors of homeomorphisms of

Barge¹,

Sánchez-Gabites²

2021

Ergod. Th. Dynam. Sys.

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In this paper we focus on compacta $K \subseteq \mathbb {R}^3$ which possess a neighbourhood basis that consists of nested solid tori $T_i$ . We call these sets toroidal. Making use of the classical notion of the geometric index of a curve inside a torus, we introduce the self-geometric index of a toroidal set K, which roughly captures how each torus $T_{i+1}$ winds inside the previous $T_i$ as $i \rightarrow +\infty $ . We then use this index to obtain some results about the realizability of toroidal sets as attractors for homeomorphisms of $\mathbb {R}^3$ .

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New Techniques for Computing Geometric Index

Cited by 5 publications

References 15 publications

Free groups as end homogeneity groups of $3$-manifolds

Free groups as end homogeneity groups of $3$-manifolds

The geometric index and attractors of homeomorphisms of $\mathbb{R}^3$

The geometric index and attractors of homeomorphisms of

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