no. 2
Continuous Stability Conditions of Type $\mathbb{A}$ and Measured Laminations of the Hyperbolic Plane
Igusa, Kiyoshi1 orcid ; Rock, Job Daisie2 orcid
1Department of Mathematics, Goldsmith 218, MS 050, Brandeis University, 415 South Street, Waltham, MA 02453, USA
2Wiskunde: Analyse, Logica en Discrete Wiskunde, Gebouw S8 (bovenste verdieping), Krijgslaan 297, B 9000 Gent, België
Annals of Representation Theory, Volume 3 (2026) no. 2, pp. 205-246

We introduce stability conditions (in the sense of King) for representable modules of continuous quivers of type $\mathbb{A}$ along with a special criteria called the four point condition. The stability conditions are defined using a generalization of $\delta $ functions, called half-$\delta $ functions. We show that for a continuous quiver of type $\mathbb{A}$ with finitely many sinks and sources, the stability conditions satisfying the four point condition are in bijection with measured laminations of the hyperbolic plane. Along the way, we extend an earlier result by the first author and Todorov regarding continuous cluster categories for linear continuous quivers of type $\mathbb{A}$ and laminations of the hyperbolic plane to all continuous quivers of type $\mathbb{A}$ with finitely many sinks and sources. We also give a formula for the continuous cluster character.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/art.40
Classification: 16G20, 13F60
Keywords: stability condition, quiver representation, continuous quiver representation, cluster category, lamination, measured lamination

Igusa, Kiyoshi  1 ; Rock, Job Daisie  2

1 Department of Mathematics, Goldsmith 218, MS 050, Brandeis University, 415 South Street, Waltham, MA 02453, USA
2 Wiskunde: Analyse, Logica en Discrete Wiskunde, Gebouw S8 (bovenste verdieping), Krijgslaan 297, B 9000 Gent, België
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
Igusa, Kiyoshi; Rock, Job Daisie. Continuous Stability Conditions of Type $\mathbb{A}$ and Measured Laminations of the Hyperbolic Plane. Annals of Representation Theory, Volume 3 (2026) no. 2, pp. 205-246. doi: 10.5802/art.40
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