[Paper Review] Relative completions of linear groups over Z[t] and Z[t,t^{-1}]

K. Knudson|ArXiv.org|Jan 21, 1998

Geometric and Algebraic Topology12 references3 citations

TL;DR

This paper computes the relative completions of SL_n(Z[t]) and SL_n(Z[t,t⁻¹]) with respect to their natural homomorphisms into SL_n(Q), generalizing Malcev completion. It provides explicit descriptions of these completions and partial computations of the rational second cohomology groups of the groups, advancing understanding of arithmetic groups over polynomial and Laurent polynomial rings over Z.

ABSTRACT

We compute the completion of the groups SL_n(Z[t]) and SL_n(Z[t,t^{-1}]) relative to the obvious homomorphisms to SL_n(Q); this is a generalization of the classical Malcev completion. We also make partial computations of the rational second cohomology of these groups.

Motivation & Objective

• To generalize the classical Malcev completion to relative completions of linear groups over Z[t] and Z[t,t⁻¹].
• To determine the structure of the completion of SL_n(Z[t]) and SL_n(Z[t,t⁻¹]) relative to the homomorphism into SL_n(Q).
• To compute partial rational second cohomology groups of these linear groups.
• To extend techniques from K-theory and group cohomology to arithmetic groups over polynomial rings.
• To provide foundational results for understanding the homotopy and cohomological properties of linear groups over Z[t] and Z[t,t⁻¹].

Proposed method

• Uses relative completion techniques in the sense of Malcev, adapted to the setting of linear groups over Z[t] and Z[t,t⁻¹].
• Applies methods from algebraic K-theory and group cohomology to analyze the structure of the completions.
• Employs spectral sequences and rational homotopy theory to study the second cohomology with rational coefficients.
• Leverages the natural inclusion of Z[t] and Z[t,t⁻¹] into Q to define the relative completion via the homomorphism into SL_n(Q).
• Analyzes the lower central series and associated graded Lie algebras to describe the completion structure.
• Utilizes results from the theory of arithmetic groups and pro-algebraic groups to derive the completion descriptions.

Experimental results

Research questions

• RQ1 What is the relative completion of SL_n(Z[t]) with respect to the homomorphism into SL_n(Q)?
• RQ2 How does the relative completion of SL_n(Z[t,t⁻¹]) differ from that of SL_n(Z[t])?
• RQ3 What is the structure of the rational second cohomology group H²(SL_n(Z[t]), Q) and H²(SL_n(Z[t,t⁻¹]), Q)?
• RQ4 To what extent can the Malcev completion be generalized to relative completions in the context of linear groups over polynomial rings?
• RQ5 What algebraic and homotopical invariants can be extracted from the relative completion of these linear groups?

Key findings

• The relative completion of SL_n(Z[t]) is described explicitly as a pro-unipotent group extension over SL_n(Q), generalizing Malcev completion.
• The relative completion of SL_n(Z[t,t⁻¹]) is shown to be isomorphic to a certain completion involving the loop group structure of Z[t,t⁻¹].
• Partial computations of the rational second cohomology groups H²(SL_n(Z[t]), Q) and H²(SL_n(Z[t,t⁻¹]), Q) are obtained, revealing non-trivial rational cohomology in certain degrees.
• The completion structures are shown to be compatible with the natural filtrations induced by the degree of polynomials in t.
• The results indicate that the relative completions capture essential arithmetic and geometric data of the linear groups over Z[t] and Z[t,t⁻¹].
• The methods provide a framework for computing completions of other arithmetic groups over similar rings.