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The mapping class group action on $\mathsf{SU}(3)$-character varieties

Published online by Cambridge University Press:  15 June 2020

WILLIAM M. GOLDMAN
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA (e-mail: [email protected])
SEAN LAWTON
Affiliation:
Department of Mathematical Sciences, George Mason University, 4400 University Drive, Fairfax, VA 22030, USA (e-mail: [email protected])
EUGENE Z. XIA
Affiliation:
Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan (e-mail: [email protected])
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Abstract

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Let $\unicode[STIX]{x1D6F4}$ be a compact orientable surface of genus $g=1$ with $n=1$ boundary component. The mapping class group $\unicode[STIX]{x1D6E4}$ of $\unicode[STIX]{x1D6F4}$ acts on the $\mathsf{SU}(3)$-character variety of $\unicode[STIX]{x1D6F4}$. We show that the action is ergodic with respect to the natural symplectic measure on the character variety.

Type
Original Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020. Published by Cambridge University Press

References

Bhosle, U. and Ramanathan, A.. Moduli of parabolic G-bundles on curves. Math. Z. 202(2) (1989), 161180.Google Scholar
Brin, M. and Stuck, G.. Introduction to Dynamical Systems. Cambridge University Press, Cambridge, 2015.Google Scholar
Duistermaat, J. J. and Kolk, J. A. C.. Lie Groups (Universitext). Springer, Berlin, 2000.Google Scholar
Florentino, C. and Lawton, S.. Topology of character varieties of Abelian groups. Topology Appl. 173 (2014), 3258.Google Scholar
Farb, B. and Margalit, D.. A Primer on Mapping Class Groups (Princeton Mathematical Series, 49). Princeton University Press, Princeton, NJ, 2012.Google Scholar
Goldman, W.. Parallelism on Lie groups and Fox’s free differential calculus. Characters in Low-Dimensional Topology (Centre de Recherches Mathématiques Proc.) (Contemporary Mathematics, 760). Eds. Collin, O., Friedl, S., Gordon, C., Tillmann, S. and Watson, L.. American Mathematical Society, Providence, RI, 2020, to appear.Google Scholar
Goldman, W. M.. The symplectic nature of fundamental groups of surfaces. Adv. Math. 54(2) (1984), 200225.Google Scholar
Goldman, W. M.. Invariant functions on Lie groups and Hamiltonian flows of surface group representations. Invent. Math. 85(2) (1986), 263302.Google Scholar
Goldman, W. M.. Ergodic theory on moduli spaces. Ann. of Math. (2) 146(3) (1997), 475507.Google Scholar
Gotô, M.. A theorem on compact semi-simple groups. J. Math. Soc. Japan 1 (1949), 270272.Google Scholar
Guillemin, V. and Pollack, A.. Differential Topology. AMS Chelsea, Providence, RI, 2010.Google Scholar
Goldman, W. M. and Xia, E. Z.. Ergodicity of mapping class group actions on SU(2)-character varieties. Geometry, Rigidity, and Group Actions (Chicago Lectures in Mathematics). University of Chicago Press, Chicago, IL, 2011, pp. 591608.Google Scholar
Huebschmann, J.. Symplectic and Poisson structures of certain moduli spaces. I. Duke Math. J. 80(3) (1995), 737756.Google Scholar
Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54). Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza.Google Scholar
Lawton, S.. $\text{SL}(3,\text{C})$ -character varieties and RP2-structures on a trinion. PhD Thesis, University of Maryland, College Park, ProQuest LLC, Ann Arbor, MI, 2006.Google Scholar
Lawton, S.. Generators, relations and symmetries in pairs of 3 × 3 unimodular matrices. J. Algebra 313(2) (2007), 782801.Google Scholar
Onishchik, A. L. and Vinberg, È. B.. Lie Groups and Algebraic Groups (Springer Series in Soviet Mathematics). Springer, Berlin, 1990. Translated from the Russian and with a preface by D. A. Leites.Google Scholar
Pickrell, D. and Xia, E. Z.. Ergodicity of mapping class group actions on representation varieties. I. Closed surfaces. Comment. Math. Helv. 77(2) (2002), 339362.Google Scholar
Pickrell, D. and Xia, E. Z.. Ergodicity of mapping class group actions on representation varieties. II. Surfaces with boundary. Transform. Groups 8(4) (2003), 397402.Google Scholar
Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York, 1982.Google Scholar