On the Surface Group Conjecture

Benjamin  Fine; Olga G. Kharlampovich; Alexei G.  Myasnikov; Vladimir N.  Remeslennikov; Gerhard Rosenberger

doi:10.71712/

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Published: 2025-08-08

DOI: https://doi.org/10.71712/

Keywords:

surface group, surface group conjecture, fully residually free group, cyclically pinched one-relator group, hyperbolic group

Benjamin Fine

Department of Mathematics, Fairfield, Fairfield University, Connecticut 06430, United States.

Olga G. Kharlampovich

Department of Mathematics and Statistics, McGill University, 805 Sherbrooke W. Montreal QC H3A 2K6, Canada.

Alexei G. Myasnikov

Department of Mathematics and Statistics, McGill University, 805 Sherbrooke W. Montreal QC H3A 2K6, Canada.

Vladimir N. Remeslennikov

Omsk Branch of the Mathematical Institute SB RAS, 13 Pevtsova Street, 64409 Omsk, Russian Federation.

Gerhard Rosenberger

Fachbereich Mathematik, Universitat Dortmund, ¨Dortmund, 44227 Dortmund, Germany.

Abstract

We consider the following conjecture. Suppose that \(G\) is a non-free non-cyclic one- relator group such that each subgroup of finite index is again aone relator group and each subgroup of infinite index is a free group. Must \(G\) be a surface group? We show that if \(G\) is a freely indecomposable fully residually free group and satisfies the property that every subgroup of infinite index is free then \(G\) is either a cyclically pinched one-relator group or a conjugacy pinched one-relator group. Further such a group \(G\) is either hyperbolic or free abelian of rank 2.

Issue

Vol. 15 (2007)

Section

Articles

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

How to Cite

Fine, B. ., Kharlampovich, O. G., Myasnikov, A. G. ., Remeslennikov, V. N. ., & Rosenberger, G. (2025). On the Surface Group Conjecture. Scientia Series A: Mathematical Sciences, 15, 1-15. https://doi.org/10.71712/