[Paper Review] The isomorphism conjecture for groups with generalized free product structure

S.K. Roushon|arXiv (Cornell University)|Oct 29, 2014

Geometric and Algebraic Topology33 references3 citations

TL;DR

This paper establishes the fibered isomorphism conjecture in K- and L-theory for groups with generalized free product structures, particularly fundamental groups of graphs of finite groups and trees of virtually cyclic groups. It proves that for residually finite groups, verifying the conjecture reduces to finitely presented one-ended groups, and confirms the conjecture for large classes of groups via group actions on trees and homological techniques.

ABSTRACT

In this article we study the K- and L-theory of groups acting on trees. We consider the problem in the context of the fibered isomorphism conjecture of Farrell and Jones. We show that in the class of residually finite groups it is enough to prove the conjecture for finitely presented groups with one end. Also, we deduce that the conjecture is true for the fundamental groups of graphs of finite groups and of trees of virtually cyclic groups. To motivate the reader we include a survey on some classical works on this subject.

Motivation & Objective

• To establish the fibered isomorphism conjecture in K- and L-theory for groups acting on trees.
• To reduce the verification of the conjecture for residually finite groups to the case of finitely presented one-ended groups.
• To prove the conjecture for fundamental groups of graphs of finite groups and of trees of virtually cyclic groups.
• To provide a foundational survey of classical results in K- and L-theory relevant to the isomorphism conjecture.
• To identify open problems and extensions, particularly for semidirect products and poly-free groups.

Proposed method

• Utilizes the fibered isomorphism conjecture framework of Farrell and Jones to analyze K- and L-theory of group rings.
• Applies homological techniques to relate the K- and L-theory of the fundamental group to those of vertex and edge stabilizers.
• Employs the theory of group actions on trees to decompose complex groups into simpler components.
• Applies results from reduced projective class groups and Whitehead groups to analyze finiteness obstructions.
• Uses the Wall finiteness obstruction and surgery theory to connect algebraic K-theory to geometric questions.
• Leverages residual finiteness and one-endedness to reduce the scope of the conjecture to finitely presented groups.

Experimental results

Research questions

• RQ1 For residually finite groups, is it sufficient to verify the fibered isomorphism conjecture for finitely presented one-ended groups?
• RQ2 Does the fibered isomorphism conjecture hold for fundamental groups of graphs of finite groups?
• RQ3 Is the conjecture true for fundamental groups of trees of virtually cyclic groups?
• RQ4 Can the fibered isomorphism conjecture be extended to semidirect products $G \rtimes \mathbb{Z}$ when $G$ is torsion-free abelian?
• RQ5 Does the conjecture hold for fundamental groups of graphs of groups with virtually cyclic edge groups, especially when the underlying graph is a tree?

Key findings

• The fibered isomorphism conjecture holds for fundamental groups of graphs of finite groups.
• The conjecture is true for fundamental groups of trees of virtually cyclic groups.
• For residually finite groups, proving the conjecture for finitely presented one-ended groups suffices.
• The conjecture is verified for groups acting on trees with finite or virtually cyclic vertex and edge stabilizers.
• The results extend to groups that are not virtually polycyclic or embeddable in Hadamard groups, including certain amalgamated products.
• The proofs remain valid in the L-theory case when using the L-theory version of [33, Theorem 4.8], as shown in recent work [6].