Ergodicity of Mapping Class Group Actions on Su(2)-character Varieties
ثبت نشده
چکیده
Let be a compact orientable surface with genus g and n boundary components ∂1, . . . , ∂n. Let b = (b1, . . . , bn) ∈ [−2, 2]n. Then the mapping class group Mod( ) acts on the relative SU(2)-character varietyXb := Homb(π , SU(2))/SU, comprising conjugacy classes of representations ρ with tr(ρ(∂i)) = bi. This action preserves a symplectic structure on the open dense smooth submanifold of Homb(π , SU(2))/SU corresponding to irreducible representations. This subset has full measure and is connected. In this note we use the symplectic geometry of this space to give a new proof that this action is ergodic.
منابع مشابه
Ergodicity of Mapping Class Group Actions on Su(2)-character Varieties
Let Σ be a compact orientable surface with genus g and n boundary components ∂1, . . . , ∂n. Let b = (b1, . . . , bn) ∈ [−2, 2]. Then the mapping class groupMod(Σ) acts on the relative SU(2)-character variety Xb := Homb(π, SU(2))/SU(2), comprising conjugacy classes of representations ρ with tr(ρ(∂i)) = bi. This action preserves a symplectic structure on the open dense smooth submanifold of Homb...
متن کاملAction of the Johnson-torelli Group on Representation Varieties
Let Σ be a compact orientable surface with genus g and n boundary components B = (B1, . . . , Bn). Let c = (c1, . . . , cn) ∈ [−2, 2]n. Then the mapping class group MCG of Σ acts on the relative SU(2)-character variety XC := HomC(π,SU(2))/SU(2), comprising conjugacy classes of representations ρ with tr(ρ(Bi)) = ci. This action preserves a symplectic structure on the smooth part of XC , and the ...
متن کاملJu n 20 00 Ergodicity of Mapping Class Group Actions on Representation Varieties , I . Closed Surfaces
We prove that the mapping class group of a closed surface acts ergodically on connected components of the representation variety corresponding to a connected compact Lie group.
متن کاملThe Algebraic Entropy of the Special Linear Character Automorphisms of a Free Group on Two Generators
In this note, we establish a connection between the dynamical degree, or algebraic entropy of a certain class of polynomial automorphisms of R3, and the maximum topological entropy of the action when restricted to compact invariant subvarieties. Indeed, when there is no cancellation of leading terms in the successive iterates of the polynomial automorphism, the two quantities are equal. In gene...
متن کاملMapping Class Actions on Moduli Spaces
It is known that the mapping class group of a compact surface S, MCG(S), acts ergodically with respect to symplectic measure on each symplectic leaf of the Poisson moduli space of flat SU(2)-bundles over S, X(S, SU(2)). In our study of how individual mapping classes act on X, we show that ergodicity does not restrict to that of cyclic subgroups of MCG(S1,1), for S1,1 a punctured torus. The acti...
متن کامل