Action of the Johnson-Torelli group on representation varieties

William M. Goldman, Eugene Zhu Xia

Research output: Contribution to journalArticle

2 Citations (Scopus)

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 languageEnglish
Pages (from-to)1449-1457
Number of pages9
JournalProceedings of the American Mathematical Society
Volume140
Issue number4
DOIs
Publication statusPublished - 2012 Apr 1

Fingerprint

Torelli Group
Dehn Twist
Character Variety
Mapping Class Group
Symplectic Structure
Conjugacy class
Null
Genus
Subgroup

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Cite this

@article{c2a336414f844e5e8bd6a2e82bd2668a,
title = "Action of the Johnson-Torelli group on representation varieties",
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.",
author = "Goldman, {William M.} and Xia, {Eugene Zhu}",
year = "2012",
month = "4",
day = "1",
doi = "10.1090/S0002-9939-2011-10972-9",
language = "English",
volume = "140",
pages = "1449--1457",
journal = "Proceedings of the American Mathematical Society",
issn = "0002-9939",
publisher = "American Mathematical Society",
number = "4",

}

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

In: Proceedings of the American Mathematical Society, Vol. 140, No. 4, 01.04.2012, p. 1449-1457.

Research output: Contribution to journalArticle

TY - JOUR

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

AU - Goldman, William M.

AU - Xia, Eugene Zhu

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 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.

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 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.

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

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 -