Nonlocal elliptic problems associated with actions of discrete groups on manifolds with boundary

TL;DR

This paper develops an index theory for elliptic operators on manifolds with boundary influenced by discrete group actions, using a Chern character in cohomology to relate symbols to fixed point data.

Contribution

It introduces a new index formula for nonlocal elliptic problems on manifolds with boundary under discrete group actions, linking operator symbols to topological invariants.

Findings

Constructed a Chern character for elliptic symbols in this setting

Derived an index formula involving fixed point manifolds

Connected operator algebra to de Rham cohomology

Abstract

Given a manifold with boundary endowed with an action of a discrete group on it, we consider the algebra of operators generated by elements in the Boutet de Monvel algebra of pseudodifferential boundary value problems and shift operators acting on functions on the manifold and its boundary. Under certain conditions on the group action, we construct a Chern character for symbols of elliptic elements in this algebra with values in a de Rham type cohomology of the fixed point manifolds and obtain an index formula in terms of this Chern character.