The Surface Group Conjectures for groups with two generators

Giles Gardam; Dawid Kielak; Alan D. Logan

The Surface Group Conjectures for groups with two generators

Giles Gardam, Dawid Kielak, Alan D. Logan

School of Engineering & Physical Sciences

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

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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
Journal	Mathematical Research Letters
Publication status	Accepted/In press - 9 Aug 2022

Keywords

math.GR
20F65 (Primary) 57M10, 20F05, 20F67, 20J05 (Secondary)

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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. ",

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AU - Gardam, Giles

AU - Kielak, Dawid

AU - Logan, Alan D.

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PY - 2022/8/9

Y1 - 2022/8/9

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 - math.GR

KW - 20F65 (Primary) 57M10, 20F05, 20F67, 20J05 (Secondary)

M3 - Article

SN - 1073-2780

JO - Mathematical Research Letters

JF - Mathematical Research Letters

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The Surface Group Conjectures for groups with two generators

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