Subgroups of Teichmuller Modular Groups |
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A₁ abelian subgroup annulus apply Lemma arational assume automorphism B₁ boundary canonical reduction system Clearly completes the proof component Q Consequently consider contains a pseudo-Anosov contradicts the fact Corollary Dehn twists denote diffeomorphism disk elements f Exposé f₁ f₂ finite index fix(e fix(ƒ fix(g fix(h fixed points foliation Frattini subgroup free group G₁ genus geodesic group G group Mods H₁ H₁(S Hence it follows homomorphism infinite cyclic intersection number invariant irreducible subgroup isotopy class length function Let F Let G linear groups maximal subgroups metric modular groups N₁ neighborhoods nilpotent nonempty nontrivial circles normal subgroup number of components obviously proof of Lemma proof of Theorem prove pseudo-Anosov element pure elements R)-polar R₁ respectively segments sequence simplex subgroup of finite subgroup of Mods sufficiently large surface Sc Theorem 3.5 Thurston transversal transverse measure trivial µ³