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

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- by William M. Goldman and Eugene Z. Xia
- Proc. Amer. Math. Soc.
**140**(2012), 1449-1457 - DOI: https://doi.org/10.1090/S0002-9939-2011-10972-9
- Published electronically: July 26, 2011
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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 $c$.## References

- William M. Goldman,
*Trace coordinates on Fricke spaces of some simple hyperbolic surfaces*, Handbook of Teichmüller theory. Vol. II, IRMA Lect. Math. Theor. Phys., vol. 13, Eur. Math. Soc., Zürich, 2009, pp. 611–684. MR**2497777**, DOI 10.4171/055-1/16 - William M. Goldman,
*Ergodic theory on moduli spaces*, Ann. of Math. (2)**146**(1997), no. 3, 475–507. MR**1491446**, DOI 10.2307/2952454 - William M. Goldman,
*The symplectic nature of fundamental groups of surfaces*, Adv. in Math.**54**(1984), no. 2, 200–225. MR**762512**, DOI 10.1016/0001-8708(84)90040-9 - Goldman, William M., Xia, Eugene Z., Ergodicity of mapping class group actions on $\mathsf {SU}(2)$-character varieties. Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992, 591–608.
- Johannes Huebschmann,
*Symplectic and Poisson structures of certain moduli spaces. I*, Duke Math. J.**80**(1995), no. 3, 737–756. MR**1370113**, DOI 10.1215/S0012-7094-95-08024-7 - Dennis Johnson,
*The structure of the Torelli group. III. The abelianization of $\scr T$*, Topology**24**(1985), no. 2, 127–144. MR**793179**, DOI 10.1016/0040-9383(85)90050-3 - Dennis Johnson,
*The structure of the Torelli group. II. A characterization of the group generated by twists on bounding curves*, Topology**24**(1985), no. 2, 113–126. MR**793178**, DOI 10.1016/0040-9383(85)90049-7 - Dennis Johnson,
*The structure of the Torelli group. I. A finite set of generators for ${\cal I}$*, Ann. of Math. (2)**118**(1983), no. 3, 423–442. MR**727699**, DOI 10.2307/2006977 - Doug Pickrell and Eugene Z. Xia,
*Ergodicity of mapping class group actions on representation varieties. II. Surfaces with boundary*, Transform. Groups**8**(2003), no. 4, 397–402. MR**2015257**, DOI 10.1007/s00031-003-0819-6 - Doug Pickrell and Eugene Z. Xia,
*Ergodicity of mapping class group actions on representation varieties. I. Closed surfaces*, Comment. Math. Helv.**77**(2002), no. 2, 339–362. MR**1915045**, DOI 10.1007/s00014-002-8343-1 - Andrew Putman,
*Cutting and pasting in the Torelli group*, Geom. Topol.**11**(2007), 829–865. MR**2302503**, DOI 10.2140/gt.2007.11.829 - van den Berg, B., On the abelianization of the Torelli group, doctoral dissertation, Universiteit Utrecht (2003).

## Bibliographic Information

**William M. Goldman**- Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
- MR Author ID: 74725
- ORCID: 0000-0002-4143-6404
- 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
- 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
- © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2869130