Pseudo-Anosov autoquivalances arising from Symplectic topology and their hyperbolic actions on stability conditions

Abstract

Within $N$-Calabi-Yau categories associated with quivers whose base graphs form trees, we delve into the study of the asymptotic behaviors of autoequivalences of a specific type. These autoequivalences, which we call "Penner type," exhibit straightforward asymptotic characteristics, making them noteworthy exemplars of "pseudo-Anosov" autoequivalences in the sense of \cite{Fan-Filip-Haiden-Katzarkov-Liu21}, and also in a stronger sense that we define in the present paper. In addition, we provide a practical methodology for calculating the stretching factors of Penner type autoequivalences. We expect that this computational approach can have applications. As an example, we establish a positive lower bound on the translation length of the induced action these autoequivalences have on the space of stability conditions. Our anticipation is that this lower bound is, in fact, exact. Notably, we have observed instances of Penner type $\Phi$ where the induced actions align precisely with this lower bound. In other words, these examples induce hyperbolic actions on the space of stability conditions.