Ziggurats and rotation numbers

Calegari, Danny; Walker, Alden

doi:10.3934/jmd.2011.5.711

Cited by 32 publications

(47 citation statements)

References 14 publications

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“…A proof using the language of rotation numbers (consistent with our notation) can be found in [2,Theorem 3.9]. However this bound is classical and was known much earlier.…”

Section: Step 3: Maximality Of the Euler Numbermentioning

confidence: 70%

A characterization of Fuchsian actions by topological rigidity

Mann

Wolff

2019

Pacific J. Math.

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We prove that any rigid representation of π1Σg in Homeo + (S 1 ) with Euler number at least g is necessarily semi-conjugate to a discrete, faithful representation into PSL(2, R). Combined with earlier work of Matsumoto, this precisely characterizes Fuchsian actions by a topological rigidity property. Though independent, this work can be read as an introduction to the companion paper [10] by the same authors.

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“…A proof using the language of rotation numbers (consistent with our notation) can be found in [2,Theorem 3.9]. However this bound is classical and was known much earlier.…”

Section: Step 3: Maximality Of the Euler Numbermentioning

confidence: 70%

A characterization of Fuchsian actions by topological rigidity

Mann

Wolff

2019

Pacific J. Math.

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“…Thus, this result is one of the several in this monograph which are completely different from (as opposed to complementary to) the other results in literature which study flexibility and rigidity of actions via rotation numbers (cf. [24,28,47,79], for instance).…”

Section: Uncountable Families Of Exotic Groupmentioning

confidence: 99%

“…In [28], Calegari and Walker study actions of the rank two free group F 2 on the circle, and the various rigidity and rationality phenomena that can be described therein. For instance, if two free generators act with rational rotation numbers r and s then the supremum Rpw, r, sq of the possible rotation numbers achieved by each fixed positive word w P F 2 in those generators is again rational.…”

Section: 75mentioning

confidence: 99%

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Flexibility of Group Actions on the Circle

Mj¹,

Koberda²,

Kim³

2019

Lecture Notes in Mathematics

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In this partly expository monograph we develop a general framework for producing uncountable families of exotic actions of certain classically studied groups acting on the circle. We show that if L is a nontrivial limit group then the nonlinear representation variety HompL, Homeo`pS 1 qq contains uncountably many semi-conjugacy classes of faithful actions on S 1 with pairwise disjoint rotation spectra (except for 0) such that each representation lifts to R. For the case of most Fuchsian groups L, we prove further that this flexibility phenomenon occurs even locally, thus complementing a result of K. Mann. We prove that each non-elementary free or surface group admits an action on S 1 that is never semi-conjugate to any action that factors through a finite-dimensional connected Lie subgroup in Homeo`pS 1 q. It is exhibited that the mapping class groups of bounded surfaces have non-semi-conjugate faithful actions on S 1 . In the process of establishing these results, we prove general combination theorems for indiscrete subgroups of PSL 2 pRq which apply to most Fuchsian groups and to all limit groups. We also show a Topological Baumslag Lemma, and general combination theorems for representations into Baire topological groups. The abundance of Z-valued subadditive defect-one quasimorphisms on these groups would follow as a corollary. We also give a mostly self-contained reconciliation of the various notions of semi-conjugacy in the extant literature by showing that they are all equivalent.

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“…For example, they are invariant under conjugation and furthermore, Jørgensen's criterion of discreteness for subgroups of PSL(2, R) [11,Theorem 2], which can be described in terms of absolute value of the trace, has an analogue for the group of real analytic diffeomorphisms of the circle (see [13,Theorem 1.2]). In this article, we give another similarity between rotation number and linear character from a viewpoint given by D. Calegari and A. Walker [5].…”

Section: Introductionmentioning

confidence: 99%

Rotation number and lifts of a Fuchsian action of the modular group on the circle

Matsuda¹

Advanced Studies in Pure Mathematics

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We characterize the semi-conjugacy class of a Fuchsian action of the modular group on the circle in terms of rotation numbers of two standard generators and that of their product. We also show that among lifts of a Fuchsian action of the modular group, only 5-fold lift admits a similar characterization. These results indicate similarity and difference between rotation number and linear character.We denote by Homeo + (S 1 ) the group of orientation-preserving homeomorphisms of the circle. We regard the circle S 1 as the quotient R/Z and denote by p : R → S 1 the projection. Let Homeo + (S 1 ) be the group of lifts of orientation-preserving homeomorphisms to R, namely, homeomorphisms of R commuting with integral translations.

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Ziggurats and rotation numbers

Cited by 32 publications

References 14 publications

A characterization of Fuchsian actions by topological rigidity

A characterization of Fuchsian actions by topological rigidity

Flexibility of Group Actions on the Circle

Rotation number and lifts of a Fuchsian action of the modular group on the circle

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