Elliptic $G$-operators on manifolds with isolated singularities

TL;DR

This paper investigates elliptic operators on manifolds with isolated singularities that are equipped with a discrete group action, establishing conditions for their Fredholm property based on principal symbols.

Contribution

It introduces a framework for analyzing elliptic operators with group actions on singular manifolds, defining a combined principal symbol and proving Fredholm criteria.

Findings

Elliptic operators have a Fredholm property characterized by their principal symbols.

The principal symbol consists of an interior symbol and a conormal symbol.

Fredholm property is established for elliptic elements in this setting.

Abstract

In the present work we study elliptic operators on manifolds with singularities in the situation, when the manifold is endowed with an action of a discrete group $G$. As usual in elliptic theory, the Fredholm property of an operator is governed by the properties of its principal symbol. We show that the principal symbol in our situation is a pair, consisting of the symbol on the main stratum (interior symbol) and the symbol at the conical point (conormal symbol). Fredholm property of elliptic elements is obtained.