Groups which are not properly realizable Louis Funar1 Francisco F Lasheras2 and Dusˇan Repovsˇ3

Louis Funar1

Description

Niveau: Supérieur, Doctorat, Bac+8
Groups which are not properly 3-realizable Louis Funar1, Francisco F. Lasheras2 and Dusˇan Repovsˇ3 ? 1Institut Fourier BP 74, UFR Mathematiques, Univ.Grenoble I 38402 Saint-Martin-d'Heres Cedex, France 2Departamento de Geometria y Topologia, Universidad de Sevilla, Apdo 1160, 41080 Sevilla, Spain 3Faculty of Mathematics and Physics, University of Ljubljana, P.O. Box 2964, Ljubljana 1001, Slovenia December 14, 2010 Abstract A group is properly 3-realizable if it is the fundamental group of a compact polyhedron whose universal covering is proper homotopically equivalent to some 3-manifold. We prove that when such a group is also quasi-simply filtered then it has pro-(finitely generated free) fundamental group at infinity and semi-stable ends. Conjecturally the quasi-simply filtration assumption is superfluous. Using these restrictions we provide the first examples of finitely presented groups which are not properly 3-realizable, for instance large families of Coxeter groups. AMS Math. Subj. Classification(2000): 57 M 50, 57 M 10, 57 M 30. Keywords and phrases: Properly 3-realizable, geometric simple connec- tivity, quasi-simple filtered group, Coxeter group. 1 Introduction The aim of this paper is to obtain necessary conditions for a finitely presented group to be properly 3-realizable, which lead conjecturally to a complete characterization.

• group


• dimensional compact


• conjecture


• group has semi-stable


• simply connected


• universal covering


• fundamental pro


• compact polyhedron


• a1 ?

Sujets

• Conjecture
• Simply connected space
• Covering space

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 23
• Langue: English

Extrait

Groupswhicharenotproperly3-realizable LouisFunar 1 ,FranciscoF.Lasheras 2 andDusˇanRepovsˇ 3∗ 1 InstitutFourierBP74,UFRMathe´matiques,Univ.GrenobleI38402Saint-Martin-d’He`resCedex,France 2 DepartamentodeGeometriayTopologia,UniversidaddeSevilla,Apdo1160,41080Sevilla,Spain 3 FacultyofMathematicsandPhysics,UniversityofLjubljana,P.O.Box2964,Ljubljana1001,Slovenia December14,2010 Abstract Agroupisproperly3-realizableifitisthefundamentalgroupofacompact polyhedronwhoseuniversalcoveringisproperhomotopicallyequivalenttosome 3-manifold.Weprovethatwhensuchagroupisalsoquasi-simplyfilteredthen ithaspro-(finitelygeneratedfree)fundamentalgroupatinfinityandsemi-stable ends.Conjecturallythequasi-simplyfiltrationassumptionissuperfluous.Using theserestrictionsweprovidethefirstexamplesoffinitelypresentedgroupswhich arenotproperly3-realizable,forinstancelargefamiliesofCoxetergroups. AMSMath.Subj.Classification(2000):57M50,57M10,57M30. Keywordsandphrases:Properly3-realizable,geometricsimpleconnec- tivity,quasi-simplefilteredgroup,Coxetergroup. 1Introduction Theaimofthispaperistoobtainnecessaryconditionsforafinitelypresented grouptobeproperly3-realizable,whichleadconjecturallytoacomplete characterization.Lasherasintroducedandstudiedthisclassofgroupsin [8,9,21].Recallthat: Definition1.1. AfinitelypresentedgroupΓissaidtobeproperly3-realizable (abbreviatedP3Rfromnowon)ifthereexistsacompact2-dimensionalpoly- hedron X withfundamentalgroupΓsuchthattheuniversalcovering X e is properhomotopyequivalenttoa3-manifold W 3 . ∗ Emails:[e-mail](L.Funar),[e-mail](F.F.Lasheras),du- [e-mail](D.Repovsˇ) 1