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The Surface Group Conjecture: Cyclically Pinched and Conjugacy Pinched One-Relator Groups

Ciobanu, L. ; Fine, B. ; Rosenberger, G.

In: Results in Mathematics, 2013, vol. 64, no. 1-2, p. 175-184

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    Titre
    • The Surface Group Conjecture: Cyclically Pinched and Conjugacy Pinched One-Relator Groups
    Auteur
    • Ciobanu, L.. Department of Mathematics, University of Neuchâtel, Rue Emile-Argand 11, 2000, Neuchâtel, Switzerland
    • Fine, B.. Fairfield University, CT, 06430, Fairfield, USA
    • Rosenberger, G.. Fachebereich Mathematik, University of Hamburg, Bundestrasse 55, 20146, Hamburg, Germany
    Type de document
    • Postprint
    Identifiant OAI-PMH
    • oai:doc.rero.ch:320397
    Summary
    The general surface group conjecture asks whether a one-relator group where every subgroup of finite index is again one-relator and every subgroup of infinite index is free (property IF) is a surface group. We resolve several related conjectures given in Fine etal. (Sci Math A 1:1-15, 2008). First we obtain the Surface Group Conjecture B for cyclically pinched and conjugacy pinched one-relator groups. That is: if G is a cyclically pinched one-relator group or conjugacy pinched one-relator group satisfying property IF then G is free, a surface group or a solvable Baumslag-Solitar Group. Further combining results in Fine etal. (Sci Math A 1:1-15, 2008) on Property IF with a theorem of Wilton (Geom Topol, 2012) and results of Stallings (Ann Math 2(88):312-334, 1968) and Kharlampovich and Myasnikov (Trans Am Math Soc 350(2):571-613, 1998) we show that Surface Group Conjecture C proposed in Fine etal. (Sci Math A 1:1-15, 2008) is true, namely: If G is a finitely generated nonfree freely indecomposable fully residually free group with property IF, then G is a surface group