[Paper Review] K- and L-theory of group rings over GL_n(Z)

Arthur Bartels, Wolfgang Lueck+2 more|arXiv (Cornell University)|Apr 11, 2012

Algebraic structures and combinatorial models17 references4 citations

TL;DR

This paper proves the K- and L-theoretic Farrell-Jones Conjecture with coefficients in additive categories for GLₙ(ℤ), establishing that the algebraic K- and L-theory of the group ring ℤ[GLₙ(ℤ)] is determined by the K- and L-theory of its virtually cyclic subgroups. The proof uses transfer reducibility, the transitivity principle, and inheritance properties under subgroups, finite products, and wreath products with finite groups.

ABSTRACT

We prove the K- and L-theoretic Farrell-Jones Conjecture (with coefficients in additive categories) for GL_n(Z).

Motivation & Objective

• To establish the K- and L-theoretic Farrell-Jones Conjecture for GLₙ(ℤ) with coefficients in additive categories.
• To extend the conjecture to arithmetic groups over number fields and groups commensurable with subgroups of GLₙ(R) for finitely generated rings R.
• To prove the conjecture with wreath products, enabling applications to colimits and hyperbolic groups.
• To demonstrate that the conjecture holds for GLₙ(ℤ) by reducing to virtually poly-cyclic groups via induction and inheritance properties.

Proposed method

• Utilize the transitivity principle to reduce the conjecture for GLₙ(ℤ) to verifying it for groups in the family (ℱₙ)≀, the wreath product closure of a specific family ℱₙ.
• Establish transfer reducibility of GLₙ(ℤ) over the family ℱₙ, which allows application of the fibered Farrell-Jones Conjecture via Theorem 5.1.
• Apply the inheritance properties of the Farrell-Jones Conjecture under subgroups, finite products, and finite index overgroups to reduce to virtually poly-cyclic groups.
• Use the fact that wreath products of virtually poly-cyclic groups with finite groups remain virtually poly-cyclic, and such groups satisfy the conjecture by prior results (Bartels-Farrell-Lück).
• Leverage the embedding of wreath products GLₙ(ℤ)≀F into GLₘ(ℤ) for some m > n to relate the standard and wreath product versions of the conjecture.
• Apply Lemma 7.1 to show the conjecture holds for GLₙ(R) and SLₙ(R) when R has finitely generated abelian group structure, using group homomorphisms to autℤ(ℤᵏ × T) and exact sequences involving finite groups.

Experimental results

Research questions

• RQ1 Does GLₙ(ℤ) satisfy the K-theoretic Farrell-Jones Conjecture with coefficients in additive categories?
• RQ2 Does GLₙ(ℤ) satisfy the L-theoretic Farrell-Jones Conjecture with coefficients in additive categories with involution?
• RQ3 Can the conjecture be extended to groups commensurable with subgroups of GLₙ(R) for finitely generated rings R?
• RQ4 Does the conjecture with wreath products hold for GLₙ(ℤ), and how does it relate to the standard conjecture?
• RQ5 Can the conjecture for GLₙ(ℤ) be reduced to the case of virtually poly-cyclic groups via induction and inheritance properties?

Key findings

• The K-theoretic and L-theoretic Farrell-Jones Conjecture hold for GLₙ(ℤ) with coefficients in additive categories, as stated in the Main Theorem.
• The conjecture extends to all groups commensurable with subgroups of GLₙ(R) for any ring R with finitely generated abelian group structure, as shown in the General Theorem.
• The conjecture with wreath products holds for GLₙ(ℤ), and this version is sufficient to deduce the standard conjecture via the transitivity principle.
• The proof relies on showing that GLₙ(ℤ) is transfer reducible over a suitable family ℱₙ, enabling application of Theorem 5.1 (ii) for the L-theoretic case.
• The conjecture holds for all hyperbolic groups with respect to the family 𝒱𝒞yc≀, due to their strong transfer reducibility and inheritance properties.
• The FJC with wreath products passes to subgroups and finite index overgroups, allowing reduction to virtually poly-cyclic groups, which satisfy the conjecture by known results.