Abstract
Motivated by the study of the autoequivalence group of triangulated categories via isometric actions on metric spaces, we consider curvature properties (CAT(0), Gromov hyperbolic) of the space of Bridgeland stability conditions with the canonical metric defined by Bridgeland. We then prove that the metric is neither CAT(0) nor hyperbolic, and the quotient metric by the natural \({\mathbb C}\)-action is not CAT(0) in case of the Kronecker quiver. Moreover, we also show the hyperbolicity of pseudo-Anosov functors defined by Dimitrov–Haiden–Katzarkov–Kontsevich, which yields the lower-bound of entropy by the translation length. Finally, pseudo-Anosov functors in case of curves have been completely classified.
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Acknowledgements
This work was partially done at The University of Sheffield which the author visited in the program “Overseas Challenge Program for Young Researchers” supported by JSPS, from September 2018 to February 2019. The author would like to express his sincere gratitude to the host Professor Tom Bridgeland for his kind hospitality and valuable discussions in Sheffield. The author would like to thank Genki Ouchi for telling him Proposition 5.4 and its proof and Atsushi Takahashi for useful comments. The author is also supported by JSPS KAKENHI Grant Number JP17J00227.
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Kikuta, K. Curvature of the space of stability conditions. manuscripta math. 171, 437–456 (2023). https://doi.org/10.1007/s00229-022-01389-9
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DOI: https://doi.org/10.1007/s00229-022-01389-9