Dynamics of the mapping class group on the moduli of a punctured sphere with rational holonomy

Joseph P. Previte, Eugene Z. Xia

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

Let M be a four-holed sphere and Γ the mapping class group of M fixing the boundary ∂M. The group Γ acts on M B(SL(2,ℂ)) = HomB+1(M), SL(2,ℂ))/SL(2,ℂ) which is the space of completely reducible SL(2,ℂ)-gauge equivalence classes of flat SL(2,ℂ)-connections on M with fixed holonomy B on ∂M. Let B∈[-2,2]4 and M B be the compact component of the real points of M B(SL(2,ℂ)). These points correspond to SU(2)-representations or SL(2,ℝ)-representations. The Γ-action preserves MB and we study the topological dynamics of the Γ-action on MB and show that for a dense set of holonomy B∈[-2, 2]4, the Γ-orbits are dense in MB. We also produce a class of representations ρ∈HomB+1(M), SL(2,ℝ)) such that the Γ-orbit of [ρ] is finite in the compact component of M B(SL(2,ℝ)), but ρ(π1(M)) is dense in SL(2, ℝ}.

Original languageEnglish
Pages (from-to)65-72
Number of pages8
JournalGeometriae Dedicata
Volume112
Issue number1
DOIs
Publication statusPublished - 2005 Apr

All Science Journal Classification (ASJC) codes

  • Geometry and Topology

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