The Surface Group Conjectures for groups with two generators

Giles Gardam; Dawid Kielak; Alan D. Logan

doi:10.4310/mrl.2023.v30.n1.a5

The Surface Group Conjectures for groups with two generators

Giles Gardam
, Dawid Kielak
, Alan D. Logan

Pure Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

The Surface Group Conjectures are statements about recognising surface groups among one-relator groups, using either the structure of their finite-index subgroups, or all subgroups. We resolve these conjectures in the two generator case. More generally, we prove that every two-generator one-relator group with every infinite-index subgroup free is itself either free or a surface group.

Original language	English
Pages (from-to)	109-123
Number of pages	15
Journal	Mathematical Research Letters
Volume	30
Issue number	1
DOIs	https://doi.org/10.4310/mrl.2023.v30.n1.a5
Publication status	Published - 21 Jun 2023

Keywords

One-relator group
Surface group

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10.4310/mrl.2023.v30.n1.a5

Gardam_2022_MRL_SGC_AAM
Copyright © 2022 International Press of Boston, Inc. All rights reserved. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://www.intlpress.com/site/pub/pages/journals/items/mrl/_home/_main/index.php
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https://arxiv.org/abs/2202.11093

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title = "The Surface Group Conjectures for groups with two generators",

abstract = "The Surface Group Conjectures are statements about recognising surface groups among one-relator groups, using either the structure of their finite-index subgroups, or all subgroups. We resolve these conjectures in the two generator case. More generally, we prove that every two-generator one-relator group with every infinite-index subgroup free is itself either free or a surface group.",

keywords = "One-relator group, Surface group",

author = "Giles Gardam and Dawid Kielak and Logan, \{Alan D.\}",

note = "Funding: This work has received funding from the European Research Council (ERC) under the European Union{\textquoteright}s Horizon 2020 research and innovation programme (Grant agreement No. 850930), and from the Engineering and Physical Sciences Research Council (EPSRC), grants EP/R035814/1 and EP/S010963/1, and was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project ID 427320536 – SFB 1442, as well as under Germany{\textquoteright}s Excellence Strategy EXC 2044–390685587, Mathematics M{\"u}nster: Dynamics–Geometry–Structure.",

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T1 - The Surface Group Conjectures for groups with two generators

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AU - Kielak, Dawid

AU - Logan, Alan D.

N1 - Funding: This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 850930), and from the Engineering and Physical Sciences Research Council (EPSRC), grants EP/R035814/1 and EP/S010963/1, and was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project ID 427320536 – SFB 1442, as well as under Germany’s Excellence Strategy EXC 2044–390685587, Mathematics Münster: Dynamics–Geometry–Structure.

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N2 - The Surface Group Conjectures are statements about recognising surface groups among one-relator groups, using either the structure of their finite-index subgroups, or all subgroups. We resolve these conjectures in the two generator case. More generally, we prove that every two-generator one-relator group with every infinite-index subgroup free is itself either free or a surface group.

AB - The Surface Group Conjectures are statements about recognising surface groups among one-relator groups, using either the structure of their finite-index subgroups, or all subgroups. We resolve these conjectures in the two generator case. More generally, we prove that every two-generator one-relator group with every infinite-index subgroup free is itself either free or a surface group.

KW - One-relator group

KW - Surface group

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VL - 30

SP - 109

EP - 123

JO - Mathematical Research Letters

JF - Mathematical Research Letters

IS - 1

ER -

The Surface Group Conjectures for groups with two generators

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Keywords

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