ACTION OF A GROTHENDIECK-TEICHMULLER GROUP ON TORSION ELEMENTS OF FULL TEICHMULLER MODULAR GROUPS OF GENUS ONE

摘要

In this paper we define a new version of Grothendieck-Teichmiiller group GR defined by three generalized equations coming from finite-order diffeomorphisms, and we prove that it is isomorphic to the known version IΓ of the Grothendieck-Teichmiiller defined in [H. Nakamura and L. Schneps, On a subgroup of the Grothendieck-Teichmiiller group acting on the tower of profinite Teichmiiller modular groups, Invent. Math. 141 (2000) 503-560]. We show that GR acts on the full mapping class groups π_1~(geom)(M_g.[n]]) for 2g — 2 + n > 0. We then prove that the conjugacy classes of prime-order torsion of π_1~(geom)(M_1, [n]) are exactly the discrete prime-order ones of π_1~(orb) (M_1,[n]). Using this we prove that GR acts on prime-order torsion elements of π_1~(geom)(M_1,[n]) in a particular way called A-conjugacy, analogous to the Galois action on inertia.