The Novikov Conjecture and Extensions of Coarsely Embeddable Groups

Abstract

This dissertation can be said to consider the Novikov conjecture for an extension of coarsely embeddable groups.

The first part of the dissertation is about defining a $C^*$-algebra associated with an extension of coarsely embeddable groups. This $C^*$-algebra comes with an action of the extension group, and we explore the properties of this action. We then construct twisted Roe algebras and twisted localization algebras associated with the extension, and develop a framework to compute their $K$-theory.

In the second part of this dissertation, we define and study the Bott map from the suspension of the localization algebra to the twisted localization algebra and the Bott map from the suspension of the Roe algebra to the twisted Roe algebra associated with the extension group. We show that the Bott map between localization algebras induces an isomorphism on $K$-theory. It follows that the strong Novikov conjecture with coefficients in any $C^*$-algebra holds for a group $G$ when a normal subgroup $N$ of $G$ and the quotient group $G/N$ are coarsely embeddable into Hilbert spaces. As a result, the group $G$ satisfies the Novikov conjecture under the same hypothesis on $N$ and $G/N$.