Relative pro-ℓ completions of mapping class groups

Journal Article

Fix a prime number ℓ. In this paper we develop the theory of relative pro-ℓ completion of discrete and profinite groups-a natural generalization of the classical notion of pro-ℓ completion-and show that the pro-ℓ completion of the Torelli group does not inject into the relative pro-ℓ completion of the corresponding mapping class group when the genus is at least 2. (See Theorem 1 below.) As an application, we prove that when g ≥ 2, the action of the pro-ℓ completion of the Torelli group Tg, 1 on the pro-ℓ fundamental group of a pointed genus g surface is not faithful. The choice of a first-order deformation of a maximally degenerate stable curve of genus g determines an action of the absolute Galois group GQ on the relative pro-ℓ completion of the corresponding mapping class group. We prove that for all g all such representations are unramified at all primes ≠ℓ when the first order deformation is suitably chosen. This proof was communicated to us by Mochizuki and Tamagawa. © 2009 Elsevier Inc.

Full Text

Duke Authors

Cited Authors

  • Hain, R; Matsumoto, M

Published Date

  • June 1, 2009

Published In

Volume / Issue

  • 321 / 11

Start / End Page

  • 3335 - 3374

Electronic International Standard Serial Number (EISSN)

  • 1090-266X

International Standard Serial Number (ISSN)

  • 0021-8693

Digital Object Identifier (DOI)

  • 10.1016/j.jalgebra.2009.02.014

Citation Source

  • Scopus