A Primer on Mapping Class Groups (PMS-49)

Farb, Benson; Margalit, Dan

doi:10.23943/princeton/9780691147949.001.0001

Cited by 317 publications

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“…Figure 3). We would like to extend this rotation to an element in MCG • (S): Figure 4 and see [11] for a definition of Dehn twists). As an element in the mapping class group MCG • (S), ρ Y acts on the set of arcs of S and thus induces a permutation on A × (S) where the tagging of ρ Y (α) at any puncture equals the tagging of the arc α at that puncture.…”

Section: Lemma 22 There Is a Canonical Bijectionmentioning

confidence: 99%

Tagged mapping class groups: Auslander–Reiten translation

Brüstle

Qiu

2015

Math. Z.

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Abstract. We give a geometric realization, the tagged rotation, of the AR-translation on the generalized cluster category associated to a surface S with marked points and non-empty boundary, which generalizes Brüstle-Zhang's result for the puncture free case.As an application, we show that the intersection of the shifts in the 3-Calabi-Yau derived category D(ΓS) associated to the surface and the corresponding Seidel-Thomas braid group of D(ΓS) is empty, unless S is a polygon with at most one puncture (i.e. of type A or D).

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Section: Lemma 22 There Is a Canonical Bijectionmentioning

confidence: 99%

Tagged mapping class groups: Auslander–Reiten translation

Brüstle

Qiu

2015

Math. Z.

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“…Rather it would be associated to the code space H Λ (Σ) of a different triangulation Λ . To remedy this, we apply a local quantum circuit that effectively implements a local geometry deformation and transforms the code 2 The MCG can also be defined using diffeomorphisms; both definitions are equivalent [23]. defined on the Λ triangulation back to the one defined on the original Λ triangulation.…”

Section: Geometric Gate Sets For Hyperbolic Turaev-viro Codesmentioning

confidence: 99%

“…A hyperbolic genus g surface can be obtained by identifying every other edge of a 4g-gon, whose angles sum to 2π, in hyperbolic space. The space of different hyperbolic metrics, Teichmüller space, corresponds to inequivalent choices of the locations of the vertices of the 4g-gon [23]. Here we consider the canonical 4g-gon, i.e.…”

Section: Dehn Twists On Genus G Surfacementioning

confidence: 99%

Universal logical gates with constant overhead: instantaneous Dehn twists for hyperbolic quantum codes

Lavasani

Zhu

Barkeshli

2019

Quantum

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A basic question in the theory of fault-tolerant quantum computation is to understand the fundamental resource costs for performing a universal logical set of gates on encoded qubits to arbitrary accuracy. Here we consider qubits encoded with constant space overhead (i.e. finite encoding rate) in the limit of arbitrarily large code distance d through the use of topological codes associated to triangulations of hyperbolic surfaces. We introduce explicit protocols to demonstrate how Dehn twists of the hyperbolic surface can be implemented on the code through constant depth unitary circuits, without increasing the space overhead. The circuit for a given Dehn twist consists of a permutation of physical qubits, followed by a constant depth local unitary circuit, where locality here is defined with respect to a hyperbolic metric that defines the code. Applying our results to the hyperbolic Fibonacci Turaev-Viro code implies the possibility of applying universal logical gate sets on encoded qubits through constant depth unitary circuits and with constant space overhead. Our circuits are inherently protected from errors as they map local operators to local operators while changing the size of their support by at most a constant factor; in the presence of noisy syndrome measurements, our results suggest the possibility of universal fault tolerant quantum computation with constant space overhead and time overhead of O(d/ log d). For quantum circuits that allow parallel gate operations, this yields the optimal scaling of spacetime overhead known to date.

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“…Fold lines are of importance in understanding the geometry of outer space, as (apart from geodesics formed by shrinking the edge of a graph) geodesics of outer space are fold lines. (An explanation of how fold lines are geodesics can be found in [FM11] or [Bes12].) Handel and Mosher proved in [HM11] that, for a fully irreducible φ, the ability to fold from a point in outer space (marked graph with lengths on edges) carrying a train track representative of a φ k to a distinct such point, depends on a decomposition of the ideal Whitehead graph for φ into local stable Whitehead graphs.…”

Section: An Ideal Whitehead Graph Definitionmentioning

confidence: 99%

Ideal Whitehead graphs in Out(Fr) III: Achieved graphs in rank 3

Pfaff

2016

J. Topol. Anal.

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Abstract. By proving precisely which singularity index lists arise from the pair of invariant foliations for a pseudo-Anosov surface homeomorphism, Masur and Smillie [MS93] determined a Teichmüller flow invariant stratification of the space of quadratic differentials. In this final paper of a three-paper series, we give a first step to an Out(Fr) analog of the Masur-Smillie theorem. Since the ideal Whitehead graphs defined by Handel and Mosher [HM11] give a strictly finer invariant in the analogous Out(Fr) setting, we determine which of the twenty-one connected, simplicial, five-vertex graphs are ideal Whitehead graphs of fully irreducible outer automorphisms in Out(F3). IntroductionLet F r denote the free group of rank r and Out(F r ) its outer automorphism group. In this paper we prove realization results for an invariant dependent only on the conjugacy class (within Out(F r )) of the outer automorphism, namely the "ideal Whitehead graph." 1.1. Main result. A "fully irreducible" (iwip) outer automorphism is the most commonly used analogue to a pseudo-Anosov mapping class and is generic. An element φ ∈ Out(F r ) is fully irreducible if no positive power φ k fixes the conjugacy class of a proper free factor of F r .We give in Subsection 2.3 the exact Out(F r ) definition of an ideal Whitehead graph. For now, to give context, we remark that, for a pseudo-Anosov surface homeomorphism, the component of an ideal Whitehead graph coming from a foliation singularity is a polygon with edges corresponding to the lamination leaf lifts bounding a principal region in the universal cover [NH86].Handel and Mosher define in [HM11] a notion of an ideal Whitehead graph for a fully irreducible outer automorphism, a finite graph whose isomorphism type is an invariant of the conjugacy class of the outer automorphism. In this paper we investigate the extent to which the Out(F r ) situation is more complicated by giving a partial answer to a question posed by Handel and Mosher in [HM11]: Question 1.1. For each r ≥ 2, which isomorphism types of graphs occur as IW(φ) for a fully irreducible φ ∈ Out(F r )? Theorem. A. Exactly eighteen of the twenty-one connected, simplicial five-vertex graphs are the ideal Whitehead graph IW(φ) for a fully irreducible outer automorphism φ ∈ Out(F 3 ).The twenty-one connected, simplicial five-vertex graphs ([CP84]

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A Primer on Mapping Class Groups (PMS-49)

Cited by 317 publications

References 0 publications

Tagged mapping class groups: Auslander–Reiten translation

Tagged mapping class groups: Auslander–Reiten translation

Universal logical gates with constant overhead: instantaneous Dehn twists for hyperbolic quantum codes

Ideal Whitehead graphs in Out(Fr) III: Achieved graphs in rank 3

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