[Paper Review] Algebraic K-theory of Geometric Groups

Gunnar Carlsson, Boris Goldfarb|arXiv (Cornell University)|May 15, 2013

Homotopy and Cohomology in Algebraic Topology44 references7 citations

TL;DR

This paper proves the Isomorphism Conjecture in algebraic K-theory for geometrically finite groups of finite asymptotic dimension over regular Noetherian rings of finite homological dimension, establishing that the integral K-theoretic assembly map is a weak equivalence. The proof combines controlled K-theory and G-theory techniques, leveraging bounded control and coarse actions to overcome limitations in standard algebraic methods.

ABSTRACT

In this paper we introduce a homotopy theoretic technique for proving that the $K$-theoretic assembly map is an equivalence. It is an extension of the methods used to prove split injectivity of the assembly and applies to any geometrically finite group. Our result is that there are two requirements which need to hold. The first is that the assembly map for the group regarded as a metric space is an equivalence. This is a non-equivariant condition and depends only on the coarse type of the word metric on the group. The second is that the group ring satisfies an algebraic coherence condition, which currently can be verified for all known groups for which the split injectivity statement for the assembly holds. The two conditions extend very broadly. In particular, both conditions hold for groups of finite asymptotic dimension. To state the main theorem precisely, given a regular Noetherian ring $A$ of finite global dimension and a group $Γ$ with finite $K(Γ,1)$ and finite asymptotic dimension, we prove that the $K$-theoretic assembly map is an equivalence. Therefore, in all dimensions the $K$-theory of $A[Γ]$ is the group homology of $Γ$ with coefficients in the $K$-theory spectrum of $A$. One of the many geometric consequences of this theorem is vanishing of the Whitehead group of $Γ$.

Motivation & Objective

• To verify the Isomorphism Conjecture in algebraic K-theory for a broad class of group rings.
• To extend Waldhausen's results on amalgamated products and HNN extensions using geometric group properties instead of combinatorial constructions.
• To establish the equivalence of the K-theoretic assembly map for group rings R[Γ] when Γ is geometrically finite and has finite asymptotic dimension.
• To resolve the failure of standard controlled algebra techniques by introducing controlled G-theory and coarse bounded actions.
• To demonstrate that the Cartan map from K-theory to G-theory is a weak equivalence under the given conditions.

Proposed method

• Use of nonconnective spectra to treat K-theory in all degrees simultaneously.
• Application of bounded control and coarse geometry to model group actions on contractible spaces.
• Introduction of controlled G-theory and the spectrum C to restore excision properties lost in standard K-theory.
• Proof that the equivariant assembly map in bounded K-theory splits under finite asymptotic dimension, implying equivalence.
• Use of Quillen’s Resolution Theorem to show that the Cartan map is a weak equivalence when R has finite homological dimension.
• Construction of a commutative diagram linking homology spectra with K⁻∞(R[Γ]) and G⁻∞(R[Γ]) to deduce the main result.

Experimental results

Research questions

• RQ1 Does the K-theoretic assembly map for R[Γ] become a weak equivalence when Γ is geometrically finite and has finite asymptotic dimension?
• RQ2 Can the Isomorphism Conjecture be proven using geometric and metric properties of groups rather than combinatorial group presentations?
• RQ3 Why do standard controlled algebra techniques fail in proving excision and localization in this context?
• RQ4 To what extent does controlled G-theory restore the necessary homotopical properties missing in K-theory?
• RQ5 Under what conditions is the Cartan map between K⁻∞(R[Γ]) and G⁻∞(R[Γ]) a weak equivalence?

Key findings

• The integral K-theoretic assembly map a(Γ,R) is a weak equivalence for geometrically finite groups Γ of finite asymptotic dimension and regular Noetherian rings R of finite homological dimension.
• The equivariant assembly map in bounded K-theory splits, implying that the G-theoretic assembly map is an equivalence under the same conditions.
• Controlled G-theory resolves the failure of excision in standard K-theory by incorporating coarse bounded actions and Grothendieck subcategories.
• The Cartan map κ: K⁻∞(R[Γ]) → G⁻∞(R[Γ]) is a weak equivalence when R has finite homological dimension and Γ is weakly regular Noetherian.
• The result implies that K₀(R[Γ]) ≅ K₀(R), so the reduced K₀-group ˜K₀(ℤ[Γ]) vanishes for such groups.
• The classical Whitehead group Wh(Γ) vanishes when the assembly map is a weak equivalence, as shown by the long exact sequence involving higher Whitehead groups.