On an equivariant version of the zeta function of a transformation

TL;DR

This paper introduces a new approach to defining an equivariant zeta function for G-invariant function germs, using Burnside rings and resolutions, extending classical concepts to orbifold contexts.

Contribution

It proposes an alternative method for defining the equivariant Lefschetz number and zeta function, and provides an A'Campo type formula for their computation.

Findings

New definition of equivariant Lefschetz number based on an alternative approach.

A formula for the equivariant monodromy zeta function in terms of resolution.

Discussion of orbifold versions of the Lefschetz number and zeta function.

Abstract

Earlier the authors offered an equivariant version of the classical monodromy zeta function of a G-invariant function germ with a finite group G as a power series with the coefficients from the Burnside ring of the group G tensored by the field of rational numbers. One of the main ingredients of the definition was the definition of the equivariant Lefschetz number of a G-equivariant transformation given by W.L\"uck and J.Rosenberg. Here we offer another approach to a definition of the equivariant Lefschetz number of a transformation and describe the corresponding notions of the equivariant zeta function. This zeta-function is a power series with the coefficients from the Burnside ring of the group G. We give an A'Campo type formula for the equivariant monodromy zeta function of a function germ in terms of a resolution. Finally we discuss orbifold versions of the Lefschetz number and of the monodromy zeta function corresponding to the two equivariant ones.