Dokument: Generalized Torelli Groups
| Titel: | Generalized Torelli Groups | |||||||
| URL für Lesezeichen: | https://docserv.uni-duesseldorf.de/servlets/DocumentServlet?id=6254 | |||||||
| URN (NBN): | urn:nbn:de:hbz:061-20071105-114732-5 | |||||||
| Kollektion: | Dissertationen | |||||||
| Sprache: | Englisch | |||||||
| Dokumententyp: | Wissenschaftliche Abschlussarbeiten » Dissertation | |||||||
| Medientyp: | Text | |||||||
| Autor: | Dr. Siegmund, Marc [Autor] | |||||||
| Dateien: |
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| Beitragende: | Prof. Dr. Grunewald, Fritz [Gutachter] Prof. Dr. Singhof, Wilhelm [Gutachter] | |||||||
| Dokumententyp (erweitert): | Dissertation | |||||||
| Dewey Dezimal-Klassifikation: | 500 Naturwissenschaften und Mathematik » 510 Mathematik | |||||||
| Beschreibung: | Let F_n be the free group on n > 1 elements and Aut(F_n) its group of automorphisms. A well-known representation of Aut(Fn) is given by rho_1 : Aut(F_n) ->> Aut(Fn/F'n) = GL(n,Z),
where F'n is the commutator subgroup of F_n. The kernel of rho_1 is called the classical Torelli group. Prof. F. Grunewald and Prof. A. Lubotzky construct more representations of finite index subgroups of Aut(F_n). By choosing a finite group G and a presentation pi: F_n ->> G they obtain an integral linear representation rho: Gamma(G,pi) -> GL(t,Z), where Gamma(G,pi) is a finite index subgroup of Aut(F_n). In this thesis I study the special case G = C_2 of this construction. The map rho, leads to the integral linear representation sigam(−1) : Gamma+(C2,pi) ->> GL(n−1,Z). Let K_n denote the kernel of sigma(−1), which fits into the following exact sequence 1 -> K_n -> Gamma+(C2,pi) ->> GL(n−1,Z) -> 1. We call the kernel K_n a generalized Torelli group. The first main theorem of this thesis states that K_n is finitely generated as a group. In the proof we give a set of generators explicitly. Note that this theorem corresponds to the famous theorem of Nielsen and Magnus, which states that the classical Torelli group is finitely generated. Further we study the abelianized group (K_n)^ab, which becomes by the exaxt sequence above a GL(n−1,Z)-module. Finally we consider higher quotients of the lower central series of K_n. Our second main theorem states the surprising fact that for i>0 the quotients of the lower central series of K_n are finite abelian groups of the form (Z/2Z)^(b_n,i). | |||||||
| Lizenz: |  Urheberrechtsschutz | |||||||
| Fachbereich / Einrichtung: | Mathematisch- Naturwissenschaftliche Fakultät » WE Mathematik » Algebra und Zahlentheorie | |||||||
| Dokument erstellt am: | 02.11.2007 | |||||||
| Dateien geändert am: | 02.11.2007 | |||||||
| Promotionsantrag am: | 09.08.2007 | |||||||
| Datum der Promotion: | 31.10.2007 |
