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Universal constraints on semigroups of hyperbolic isometries
Argyris Christodoulou
The Open University
Universal constraints on semigroups of hyperbolic isometries - - PowerPoint PPT Presentation
Universal constraints on semigroups of hyperbolic isometries - - PowerPoint PPT Presentation
Universal constraints on semigroups of hyperbolic isometries Argyris Christodoulou The Open University CAFT 2018 Problem and Motivation Determine all the finite collections of hyperbolic isometries f 1 f 2 f n
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Problem and Motivation
Determine all the finite collections of hyperbolic isometries f1❀ f2❀ ✿ ✿ ✿ ❀ fn for which the semigroup ❤f1❀ f2❀ ✿ ✿ ✿ ❀ fn✐ satisfies certain discreteness properties.
- A. Avila, J. Bochi and J.-C. Yoccoz, Uniformly hyperbolic finite-valued
SL(2❀ ❘)-cocycles, Comment. Math. Helv. 85 (2010), no. 4, 813–884.
- M. Jacques, I. Short, Dynamics of hyperbolic isometries, available at
https://arxiv.org/abs/1609.00576.
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Hyperbolic geometry
(❉❀ ✚)
2❥dz❥ 1❥z❥2
✼✦
✒
- ❀
✒ ✷ ❘❀ ✷ ❉✿
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Hyperbolic geometry
(❉❀ ✚)
2❥dz❥ 1❥z❥2
Isometries of the hyperbolic plane: z ✼✦ ei✒ z z0 1 z0z ❀ where ✒ ✷ ❘❀ z0 ✷ ❉✿
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Classification of M¨
- bius transformations
elliptic parabolic hyperbolic
- ne fixed point
- ne fixed point
two fixed points inside
- n the boundary
- n the boundary
✚ ❀
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Classification of M¨
- bius transformations
elliptic parabolic hyperbolic
- ne fixed point
- ne fixed point
two fixed points inside
- n the boundary
- n the boundary
✚ ❀
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Classification of M¨
- bius transformations
elliptic parabolic hyperbolic
- ne fixed point
- ne fixed point
two fixed points inside
- n the boundary
- n the boundary
Definition
The translation length of a hyperbolic transformation f is the distance ✚(f (w)❀ w), for any point w on the axis of f .
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Semigroups of M¨
- bius transformations
Definition
In this talk, a semigroup is a collection of M¨
- bius transformations that is
closed under composition.
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Semigroups of M¨
- bius transformations
Definition
In this talk, a semigroup is a collection of M¨
- bius transformations that is
closed under composition.
Definition
A semigroup is said to be discrete if it has no accumulation points in the M¨
- bius group.
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Semigroups of M¨
- bius transformations
Definition
In this talk, a semigroup is a collection of M¨
- bius transformations that is
closed under composition.
Definition
A semigroup is said to be discrete if it has no accumulation points in the M¨
- bius group.
Definition
A semigroup S is called semidiscrete if the identity is not an accumulation point of S.
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Semigroups of M¨
- bius transformations
Definition
In this talk, a semigroup is a collection of M¨
- bius transformations that is
closed under composition.
Definition
A semigroup is said to be discrete if it has no accumulation points in the M¨
- bius group.
Definition
A semigroup S is called semidiscrete if the identity is not an accumulation point of S.
- Example. A semidiscrete semigroup that is not discrete.
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Finitely-generated Semigroups
Definition
A semigroup S is called finitely-generated if there exists a finite collection
- f M¨
- bius transformations ❋ such that every element of S can be written
as a composition of elements of ❋. ❋ ❉
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Finitely-generated Semigroups
Definition
A semigroup S is called finitely-generated if there exists a finite collection
- f M¨
- bius transformations ❋ such that every element of S can be written
as a composition of elements of ❋. The transformations in ❋ are called the generators of S. ❉
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Finitely-generated Semigroups
Definition
A semigroup S is called finitely-generated if there exists a finite collection
- f M¨
- bius transformations ❋ such that every element of S can be written
as a composition of elements of ❋. The transformations in ❋ are called the generators of S.
Theorem (Jacques–Short, 2017)
Let S be a finitely-generated semigroup. If there exists a non-trivial closed subset X of ❉ that is mapped strictly inside itself by each generator, then S is semidiscrete.
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Examples
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Examples
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Examples
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Examples
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Examples
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Examples
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Examples
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Examples
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Examples
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Examples
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Examples
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Examples
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Exceptional cases
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Exceptional cases
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Exceptional cases
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Exceptional cases
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Exceptional cases
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Exceptional cases
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Constraints on the translation lengths
Theorem
Suppose that S is a semigroup generated by the hyperbolic transformations f1❀ f2❀ ✿ ✿ ✿ ❀ fn, and let ✜i be the translation length of fi. There exist ✧ ❃ 0 and M ❃ 0 such that: ✭✐✮ if ✜i ❃ M, for all i ✷ ❢1❀ ✿ ✿ ✿ ❀ n❣, then S is semidiscrete, ✭✐✐✮ if ✜j❀ ✜k ❁ ✧, for some j❀ k ✷ ❢1❀ ✿ ✿ ✿ ❀ n❣, then S is not semidiscrete. ✧ ✒
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Constraints on the translation lengths
Theorem
Suppose that S is a semigroup generated by the hyperbolic transformations f1❀ f2❀ ✿ ✿ ✿ ❀ fn, and let ✜i be the translation length of fi. There exist ✧ ❃ 0 and M ❃ 0 such that: ✭✐✮ if ✜i ❃ M, for all i ✷ ❢1❀ ✿ ✿ ✿ ❀ n❣, then S is semidiscrete, ✭✐✐✮ if ✜j❀ ✜k ❁ ✧, for some j❀ k ✷ ❢1❀ ✿ ✿ ✿ ❀ n❣, then S is not semidiscrete. The numbers ✧ and M depend only on the geometric configuration of the axes of the generators. d ✒
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Scetch of proof
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Sketch of proof
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Sketch of proof
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Sketch of proof
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Sketch of proof
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Future work
✎ What happens if ✧ ❁ ✜i ❁ M? ✎ ✜ ❃ ❀ ✿ ✿ ✿ ❀
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Future work
✎ What happens if ✧ ❁ ✜i ❁ M? ✎ Discrete semigroups ✜ ❃ ❀ ✿ ✿ ✿ ❀
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Future work
✎ What happens if ✧ ❁ ✜i ❁ M? ✎ Discrete semigroups ✘ ✜ ❃ ❀ ✿ ✿ ✿ ❀
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