Relative Pro-$\ell$ Completions of Mapping Class Groups
| dc.creator | Hain, Richard | |
| dc.creator | Matsumoto, Makoto | |
| dc.date | 2008-02-06 | |
| dc.date | 2009-02-18 | |
| dc.date.accessioned | 2026-07-07T12:42:28Z | |
| dc.date.available | 2026-07-07T12:42:28Z | |
| dc.description | Fix a prime number ell. In this paper we develop the theory of relative pro-ell completion of discrete and profinite groups -- a natural generalization of the classical notion of pro-ell completion -- and show that the pro-ell completion of the Torelli group does not inject into the relative pro-ell completion of the corresponding mapping class group when the genus is at least 3. As an application, we prove that when g > 2, the action of the pro-ell completion of the Torelli group T_{g,1} on the pro-ell 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 G_Q on the relative pro-ell completion of the corresponding mapping class group. We prove that for all g all such representations are unramified at all primes \neq ell when the first order deformation is suitably chosen. This proof was communicated to us by Mochizuki and Tamagawa. | |
| dc.description | A few minor changes. Will appear in Lehrer volume of J. Algebra | |
| dc.identifier | https://arxiv.org/abs/0802.0806 | |
| dc.identifier | http://arxiv.org/abs/0802.0806 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/220085 | |
| dc.subject | Number Theory | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Geometric Topology | |
| dc.title | Relative Pro-$\ell$ Completions of Mapping Class Groups | |
| dc.type | text |