Relative Ends, l^2 Invariants and Property (T)

TL;DR

This paper proves new splitting theorems for groups with specific properties, linking l^2 invariants, property (T), and subgroup structures, advancing understanding of group decompositions and conjectures.

Contribution

It establishes splitting theorems for groups based on l^2 invariants and property (T), and verifies the Kropholler Conjecture under certain conditions.

Findings

Splitting theorems for groups with non-trivial l^2 Betti number.

Verification of the Kropholler Conjecture for specific subgroup pairs.

Groups with property (T) and Poincare duality split over certain subgroups.

Abstract

We establish a splitting theorem for one-ended groups H<G such that \tilde{e}(G;H)> 2 and the almost malnormal closure of H is a proper subgroup of G. This yields splitting theorems for groups G with non-trivial first l^2 Betti number (\beta^2_1(G)). We verify the Kropholler Conjecture for pairs H < G satisfying \beta^2_1(G) > \beta^2_1(H). We also prove that every n-dimensional Poincare duality (PD^n) group containing a PD^(n-1) group H with property (T) splits over a subgroup commensurable with H.