Smooth Version of Johnson's Problem Concerning Derivations of Group Algebras

TL;DR

This paper provides a smooth analogue of Johnson's problem by characterizing the outer derivations of a group algebra through the cohomology of the Cayley complex, linking algebraic derivations to topological invariants.

Contribution

It introduces a smooth version of Johnson's problem and describes the algebra of outer derivations in terms of the Cayley complex's cohomology.

Findings

Outer derivations are isomorphic to the first cohomology group with compact supports.

The algebra of outer derivations is described via the Cayley complex.

The approach connects algebraic derivations with topological invariants.

Abstract

A description of the algebra of outer derivations of a group algebra of a finitely presented discrete group is given in terms of the Cayley complex of the groupoid of the adjoint action of the group. This task is a smooth version of Johnson's problem concerning the derivations of a group algebra. It is shown that the algebra of outer derivations is isomorphic to the group of the one-dimensional cohomology with compact supports of the Cayley complex over the field of complex numbers.