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

作者：Collas Benjamin^*

来源：International Journal of Number Theory, 2012, 8(3): 763-787.

DOI：10.1142/S1793042112500431

• Moduli spaces of curves
• mapping class groups
• elliptic curves

In this paper we define a new version of Grothendieck-Teichmuller group (GR) over cap defined by three generalized equations coming from finite-order diffeomorphisms, and we prove that it is isomorphic to the known version Gamma of the Grothendieck-Teichmuller defined in [H. Nakamura and L. Schneps, On a subgroup of the Grothendieck-Teichmuller group acting on the tower of profinite Teichmuller modular groups, Invent. Math. 141 (2000) 503-560]. We show that (GR) over cap acts on the full mapping class groups pi(geom)(1) (M-g,M-[n]) for 2g - 2 + n > 0. We then prove that the conjugacy classes of prime-order torsion of pi(geom)(1) (M-1,M-[n]) are exactly the discrete prime-order ones of pi(orb)(1) (M-1,M-[n]). Using this we prove that (GR) over cap acts on prime-order torsion elements of pi(geom)(1) (M-1,M-[n]) in a particular way called lambda-conjugacy, analogous to the Galois action on inertia.

• 出版日期2012-5