### Abstract

Let Σ be a compact orientable surface with genus g and n boundary components B = (B1, ..., Bn). 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 XC, 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.

Original language | English |
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Pages (from-to) | 1449-1457 |

Number of pages | 9 |

Journal | Proceedings of the American Mathematical Society |

Volume | 140 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2012 Apr 1 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Proceedings of the American Mathematical Society*,

*140*(4), 1449-1457. https://doi.org/10.1090/S0002-9939-2011-10972-9

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*Proceedings of the American Mathematical Society*, vol. 140, no. 4, pp. 1449-1457. https://doi.org/10.1090/S0002-9939-2011-10972-9

**Action of the Johnson-Torelli group on representation varieties.** / Goldman, William M.; Xia, Eugene Z.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Action of the Johnson-Torelli group on representation varieties

AU - Goldman, William M.

AU - Xia, Eugene Z.

PY - 2012/4/1

Y1 - 2012/4/1

N2 - Let Σ be a compact orientable surface with genus g and n boundary components B = (B1, ..., Bn). 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 CHom 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 XC, 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.

AB - Let Σ be a compact orientable surface with genus g and n boundary components B = (B1, ..., Bn). 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 CHom 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 XC, 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.

UR - http://www.scopus.com/inward/record.url?scp=84856989385&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84856989385&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-2011-10972-9

DO - 10.1090/S0002-9939-2011-10972-9

M3 - Article

AN - SCOPUS:84856989385

VL - 140

SP - 1449

EP - 1457

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 4

ER -