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

Published electronically:
July 26, 2011

MathSciNet review:
2869130

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a compact orientable surface with genus and boundary components . Let . Then the mapping class group of acts on the relative -character variety , comprising conjugacy classes of representations with . This action preserves a symplectic structure on the smooth part of , and the corresponding measure is finite. Suppose and . Let be the subgroup generated by Dehn twists along null homologous simple loops in . Then the action of on is ergodic for almost all .

**1.**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**, 10.4171/055-1/16**2.**William M. Goldman,*Ergodic theory on moduli spaces*, Ann. of Math. (2)**146**(1997), no. 3, 475–507. MR**1491446**, 10.2307/2952454**3.**William M. Goldman,*The symplectic nature of fundamental groups of surfaces*, Adv. in Math.**54**(1984), no. 2, 200–225. MR**762512**, 10.1016/0001-8708(84)90040-9**4.**Goldman, William M., Xia, Eugene Z., Ergodicity of mapping class group actions on -character varieties. Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992, 591-608.**5.**Johannes Huebschmann,*Symplectic and Poisson structures of certain moduli spaces. I*, Duke Math. J.**80**(1995), no. 3, 737–756. MR**1370113**, 10.1215/S0012-7094-95-08024-7**6.**Dennis Johnson,*The structure of the Torelli group. III. The abelianization of 𝒯*, Topology**24**(1985), no. 2, 127–144. MR**793179**, 10.1016/0040-9383(85)90050-3**7.**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**, 10.1016/0040-9383(85)90049-7**8.**Dennis Johnson,*The structure of the Torelli group. I. A finite set of generators for \cal𝐼*, Ann. of Math. (2)**118**(1983), no. 3, 423–442. MR**727699**, 10.2307/2006977**9.**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**, 10.1007/s00031-003-0819-6**10.**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**, 10.1007/s00014-002-8343-1**11.**Andrew Putman,*Cutting and pasting in the Torelli group*, Geom. Topol.**11**(2007), 829–865. MR**2302503**, 10.2140/gt.2007.11.829**12.**van den Berg, B., On the abelianization of the Torelli group, doctoral dissertation, Universiteit Utrecht (2003).

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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:
http://dx.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.