Huber's theorem for hyperbolic orbisurfaces

Isospectral Finiteness of Hyperbolic Orbisurfaces

We discuss questions of isospectrality for hyperbolic orbisurfaces, examining the relationship between the geometry of an orbisurface and its Laplace spectrum. We show that certain hyperbolic orbisurfaces cannot be isospectral, where the obstructions involve the number of singular points and genera of our orbisurfaces. Using a version of the Selberg Trace Formula for hyperbolic orbisurfaces, we show that the Laplace spectrum determines the length spectrum and the orders of the singular points, up to finitely many possibilities. Conversely, knowledge of the length spectrum and the orders of the…

扭曲塞尔伯格zeta函数的除数

The divisor of the twisted Selberg zeta function

研究具有有限维酉表示扭曲的双曲轨形面的塞尔伯格zeta函数，通过拉普拉斯共振等建立因式分解公式，推广前人结果，解释其零点和极点，轨形奇点在未扭曲时也对公式有新贡献。

紧致双曲轨道曲面上的扭曲 Ruelle zeta 函数与 Reidemeister-Turaev 挠率

The twisted Ruelle zeta function on compact hyperbolic orbisurfaces and Reidemeister-Turaev torsion

该文研究紧致双曲轨道曲面上扭曲 Ruelle zeta 函数在零点取值与 Reidemeister-Turaev 挠率的关系，推广 Fried 猜想至非酉表示并解决相位符号问题，同时计算消失阶与首项系数。

Isospectral hyperbolic surfaces have matching geodesics

We show that if two closed hyperbolic surfaces (not necessarily orientable or even connected) have the same Laplace spectrum, then for every length they have the same number of orientation-preserving geodesics and the same number of orientation-reversing geodesics. Restricted to orientable surfaces, this result reduces to Huber's theorem of 1959. Appropriately generalized, it extends to hyperbolic 2-orbifolds (possibly disconnected). We give examples showing that it fails for disconnected flat 2-orbifolds.