[Paper Review] Commutator width in Chevalley groups

Roozbeh Hazrat, A. V. Stepanov+2 more|arXiv (Cornell University)|Jun 11, 2012

Finite Group Theory Research77 references24 citations

TL;DR

This paper investigates commutator width in Chevalley groups, demonstrating that elementary Chevalley groups over most rings have infinite commutator width due to sparse commutator sets. Using localization techniques and relative commutator formulas, the authors show that commutators in elementary generators have finite width, but full groups rarely achieve bounded commutator width, especially over rings of dimension ≥2. The key contribution is a structural explanation for the rarity of finite commutator width in these groups.

ABSTRACT

The present paper is the [slightly expanded] text of our talk at the Conference "Advances in Group Theory and Applications" at Porto Cesareo in June 2011. Our main results assert that [elementary] Chevalley groups very rarely have finite commutator width. The reason is that they have very few commutators, in fact, commutators have finite width in elementary generators. We discuss also the background, bounded elementary generation, methods of proof, relative analogues of these results, some positive results, and possible generalisations.

Motivation & Objective

• Understand why elementary Chevalley groups typically lack finite commutator width over most rings.
• Establish that commutators in elementary generators have finite width, despite the full group having infinite commutator width.
• Investigate the role of ring dimension and arithmetic structure (e.g., Dedekind rings) in determining commutator width.
• Explore relative and mixed commutator subgroups to generalize results beyond absolute groups.
• Provide a framework using localization and commutator calculus to analyze bounded generation and related properties.

Proposed method

• Apply localization techniques to reduce global group-theoretic problems to local ones, enabling inductive arguments.
• Use relative commutator formulas and unitriangular factorizations to analyze the structure of elementary subgroups.
• Employ conjugation calculus and commutator identities to control the length of products in elementary generators.
• Analyze bounded generation in terms of elementary transvections and their products, especially in SL(n, R) and related groups.
• Utilize the structure of Chevalley groups over rings with higher Krull dimension to show that commutator sets are too sparse for finite width.
• Apply results from algebraic K-theory and representation theory to relate commutator width to properties like Kazhdan's property (T).

Experimental results

Research questions

• RQ1 Under what conditions do elementary Chevalley groups have finite commutator width?
• RQ2 Why do most Chevalley groups fail to have finite commutator width, despite having finite width in elementary generators?
• RQ3 How does the Krull dimension of the underlying ring affect commutator width in Chevalley groups?
• RQ4 What is the role of relative and mixed commutator subgroups in bounded generation and commutator width?
• RQ5 Can bounded generation in elementary generators imply Kazhdan's property (T), and under what ring conditions?

Key findings

• Elementary Chevalley groups over rings of Krull dimension ≥2 generally do not have finite commutator width.
• Commutators in elementary generators have finite width, even though the full group may not.
• Over Dedekind rings of arithmetic type with infinite multiplicative group, elements of Ead(Φ, R) are products of at most 3 commutators.
• SL(n, Z) for n ≥3 is conjectured to have commutator width 2, but no general bound better than 4 is known.
• SL(n, Z[x]) for n ≥3 may not have bounded width with respect to elementary generators, and this would imply Kazhdan's property (T).
• SL(2, Z) admits polynomial parametrization with 46 parameters, using both elementary and pre-stability kernel generators, indicating bounded generation in a non-standard generating set.