TY - JOUR

T1 - Dynamics of the mapping class group on the moduli of a punctured sphere with rational holonomy

AU - Previte, Joseph P.

AU - Xia, Eugene Z.

N1 - Funding Information:
Xia gratefully acknowledges partial support by the National Science Council Taiwan grants NSC 91-2115-M-006-022 and 92-2115-M-006-008.

PY - 2005/4

Y1 - 2005/4

N2 - 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+(π1(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+(π1(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}.

AB - 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+(π1(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+(π1(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}.

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U2 - 10.1007/s10711-004-5106-8

DO - 10.1007/s10711-004-5106-8

M3 - Article

AN - SCOPUS:23944452652

SN - 0046-5755

VL - 112

SP - 65

EP - 72

JO - Geometriae Dedicata

JF - Geometriae Dedicata

IS - 1

ER -