The isomorphism conjecture in L-theory: graphs of groups

TL;DR

This paper investigates the Fibered Isomorphism Conjecture in L-theory for groups acting on trees, proving it for specific classes like wreath products of abelian groups and free metabelian groups, and extending results to pseudoisotopy theory.

Contribution

It establishes the conjecture for new classes of groups and extends L-theory results, advancing understanding of the isomorphism conjecture in geometric group theory.

Findings

Proved the conjecture for wreath products of abelian groups

Confirmed the conjecture for free metabelian groups

Extended results to pseudoisotopy theory

Abstract

We study the Fibered Isomorphism Conjecture of Farrell and Jones in L-theory for groups acting on trees. In several cases we prove the conjecture. This includes wreath products of abelian groups and free metabelian groups. We also deduce the conjecture in pseudoisotopy theory for these groups. Finally in B of Theorem 1.1 we prove the L-theory version of [[7], Theorem 1.2].