Poincar\'e series and Coxeter functors for Fuchsian singularities

TL;DR

This paper derives a formula for the Poincaré series of Fuchsian singularities using Coxeter elements and provides geometric interpretations by lifting algebraic structures to triangulated categories.

Contribution

It introduces a conceptual approach to compute Poincaré series for Fuchsian singularities via Coxeter elements and Eichler-Siegel transformations, with geometric insights.

Findings

Derived a formula for Poincaré series of Fuchsian singularities.

Connected algebraic structures to geometric interpretations in triangulated categories.

Utilized Coxeter elements and Eichler-Siegel transformations in the analysis.

Abstract

We consider Fuchsian singularities of arbitrary genus and prove, in a conceptual manner, a formula for their Poincar\'e series. This uses Coxeter elements involving Eichler-Siegel transformations. We give geometrical interpretations for the lattices and isometries involved, lifting them to triangulated categories.