## Subgroups of Teichmuller Modular Groups |

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abelian action addition apply assertion assume automorphism boundary bounded canonical circle Clearly closed compact component Q condition connected Consequently consider consists construction contains contradicts Corollary corresponding cyclic defined definition denote diffeomorphism ends equal example exists Exposé extended fact finite index fix(e fixed foliation follows function genus geodesic homomorphism identity implies induces infinite intersection invariant irreducible isotopy class least Lemma length length function Let G lies limit linear maximal subgroups metric Mods modular groups Moreover natural neighborhoods nontrivial normal obtain obviously orientation particular points proof proof of Theorem properties prove pseudo-Anosov element recall reduction system replace represented respectively role satisfied segments sequence singularities space subgroup of finite subgroup of Mods sufficiently large Suppose surface Theorem Theorem 3.5 theory transversal trivial