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OP-10374Status unassessedGeometry and topology

Genus 1 groups.

Configuration spaces of graphs · AIMPL workshop record

MSC 55 · Algebraic topology

Editorial mapping of AIMPL list topic: Configuration spaces of graphs. http://aimpl.org/configgraph/ Not an author-supplied MSC code.

Workshop record · status unassessedAs of 2026-09-10 source snapshot; not a current-status certificationSource & evidence ↓
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Problem statement

Is the 2-string braid group of a non-planar graph genus 1 and how can we tell if a group is genus 1 or not?

Ko and Park showed that for if the 3 string braid group of a graph is genus 1, then the graph doesn't contain Θ4\Theta_4 as a topological minor. They conjecture that if Γ\Gamma doesn't contain Θ4\Theta_4 as topological minor, then the nn-string braid group of Γ\Gamma is genus 1.

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Background

Question context

A genus 1 group is one where every relator is a commutator. They sit somewhere between right-angled artin groups and commutator-relator groups.

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References
  1. [1]Configuration spaces of graphs · Original question link
  2. [2]AIMPL source bibliography link
  3. [3]AIMPL text license · CC BY-SA 3.0 linkAIM Problem Lists (AIMPL), original contributors; CC BY-SA 3.0. Adaptation: structural TeX converted to Markdown, documented source macros expanded, and source bibliography links added. Mathematical wording retained. These adapted source texts remain under CC BY-SA 3.0.

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