Ergodicity of mapping class group actions on representation varieties, II. Surfaces with boundary

Pickrell, Doug; Xia, Eugene Z.

doi:10.1007/s00031-003-0819-6

Cited by 14 publications

(11 citation statements)

References 7 publications

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“…The ergodicity of the MCG-action on X C (SU(2)) was proved in [2,4]. See [10,9] for similar results when G is a general compact group. Here we prove the following ergodicity result: Theorem 1.1.…”

Section: Introductionmentioning

confidence: 87%

Action of the Johnson-Torelli group on representation varieties

Goldman

Xia

2012

Proc. Amer. Math. Soc.

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Let Σ be a compact orientable surface with genus g and n boundary components B = (B 1 , . . . , B n ). Let c = (c 1 , . . . , c n ) ∈ [−2, 2] n . Then the mapping class group MCG of Σ acts on the relative SU(2)-character variety X C := Hom C (π, SU(2))/SU (2), comprising conjugacy classes of representations ρ with tr(ρ(B i )) = c i . This action preserves a symplectic structure on the smooth part of X C , and the corresponding measure is finite. Suppose g = 1 and n = 2. Let J ⊂ MCG be the subgroup generated by Dehn twists along null homologous simple loops in Σ. Then the action of J on X C is ergodic for almost all c.

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Section: Introductionmentioning

confidence: 87%

Action of the Johnson-Torelli group on representation varieties

Goldman

Xia

2012

Proc. Amer. Math. Soc.

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“…The first ingredient is the ergodicity of Mod(S) on the components Rep(π, G) τ as in [17,46,47,25]. Indeed, as noted in [17], the formal property Hom(π, G × G) = Hom(π, G) × Hom(π, G), The proof crucially uses the multiplier criterion for weak mixing, as in in Glasner-Weiss [15]: the diagonal action on a Cartesian product of any ergodic probability space with a weakly mixing probability space is ergodic.…”

Section: Connected Components Of Representation Varietiesmentioning

confidence: 99%

Mixing Flows on Moduli Spaces of Flat Bundles over Surfaces

Forni¹,

Goldman²

2018

Geometry and Physics: Volume II

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We extend Teichmüller dynamics to a flow on the total space of a flat bundle of deformation spaces Rep(π, G) of representations of the fundamental group π of a fixed surface S. The resulting dynamical system is a continuous version of the action of the mapping class group Mod(S) of S on Rep(π, G). We observe how ergodic properties of the Mod(S)-action relate to this flow. When G is compact, this flow is strongly mixing over each component of Rep(π, G) and of each stratum of the Teichmüller unit sphere bundle over the Riemann moduli space M(S). We prove ergodicity for the analogous lift of the Weil-Petersson geodesic local. flow.Date: August 1, 2017. 2010 Mathematics Subject Classification. Primary: 58D27 Moduli problems for differential geometric structures; Secondary: 37A99 Ergodic Theory, 57M50 Geometric structures on low-dimensional manifolds, 22F50, Groups as automorphisms of other structures.

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“…By work of Goldman [Gol07,Gol97], Gelander [Gel08], the action of Aut(F n ) on Hom(F n , G) and the action of Out(F n ) on Hom(F n , G)/G are both ergodic for n ≥ 3. Furthermore, Pickrell and Xia [PX02,PX03], based on Goldman's results, showed that the action of Γ on M is ergodic when Σ is closed. When Σ has boundary, the mapping class group preserves the subsets of M defined by requiring a representation ̺ : π 1 (Σ) → G to map each boundary component into a prescribed conjugacy class in G; the action of Γ on each such subset is ergodic.…”

Section: Introductionmentioning

confidence: 99%