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Action of the Johnson-Torelli group on representation varieties


Authors: William M. Goldman and Eugene Z. Xia
Journal: Proc. Amer. Math. Soc. 140 (2012), 1449-1457
MSC (2010): Primary 57M05, 22D40, 13P10
DOI: https://doi.org/10.1090/S0002-9939-2011-10972-9
Published electronically: July 26, 2011
MathSciNet review: 2869130
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Abstract: Let $ \Sigma $ be a compact orientable surface with genus $ g$ and $ n$ boundary components $ B = (B_1,\dots , B_n)$. Let $ c = (c_1,\dots ,c_n)\in [-2,2]^n$. Then the mapping class group $ \mathsf {MCG}$ of $ \Sigma $ acts on the relative $ \mathsf {SU}(2)$-character variety $ \mathfrak{X}_{\mathcal {C}}:=\mathsf {Hom}_\mathcal {C}(\pi ,\mathsf {SU}(2))/\mathsf {SU}(2)$, comprising conjugacy classes of representations $ \rho $ with $ \mathfrak{tr}(\rho (B_i)) = c_i$. This action preserves a symplectic structure on the smooth part of $ \mathfrak{X}_{\mathcal {C}}$, and the corresponding measure is finite. Suppose $ g =1$ and $ n = 2$. Let $ \mathcal {J} \subset \mathsf {MCG}$ be the subgroup generated by Dehn twists along null homologous simple loops in $ \Sigma $. Then the action of $ \mathcal {J}$ on $ \mathfrak{X}_{\mathcal {C}}$ is ergodic for almost all $ c$.


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Additional Information

William M. Goldman
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: wmg@math.umd.edu

Eugene Z. Xia
Affiliation: Department of Mathematics, National Center for Theoretical Sciences, National Cheng-kung University, Tainan 701, Taiwan
Email: ezxia@ncku.edu.tw

DOI: https://doi.org/10.1090/S0002-9939-2011-10972-9
Received by editor(s): April 26, 2010
Received by editor(s) in revised form: December 24, 2010
Published electronically: July 26, 2011
Additional Notes: The first author gratefully acknowledges partial support by National Science Foundation grant DMS070781.
The second author gratefully acknowledges partial support by the National Science Council, Taiwan, with grants 96-2115-M-006-002 and 97-2115-M-006-001-MY3.
Communicated by: Bryna Kra
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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