[Paper Review] Rapid decay and Baum-Connes for large type Artin groups

Laura Ciobanu, Derek F. Holt+1 more|arXiv (Cornell University)|Mar 6, 2012

Geometric and Algebraic Topology14 references4 citations

TL;DR

This paper establishes the rapid decay (RD) property for Artin groups of large type under specific combinatorial conditions, particularly excluding triangles with labels (3,3,m), and proves that such groups satisfy the Baum-Connes conjecture without coefficients when certain structural constraints hold—such as being 3-generated, triangle-free, or admitting a suitable edge orientation. The results extend RD and Baum-Connes validity to all extra-large type Artin groups and many large type groups.

ABSTRACT

We prove that many Artin groups of large type satisfy the rapid decay property, including all those of extra-large type. For many of these, including all 3-generator groups of extra-large type, a result of Lafforgue applies to show that the groups satisfy the Baum-Connes conjecture without coefficients. Our proof of rapid decay combines elementary analysis with combinatorial techniques, and relies on properties of geodesic words in Artin groups of large type that were observed in an earlier publication by two of the authors of this current article.

Motivation & Objective

• To establish the rapid decay (RD) property for Artin groups of large type that avoid certain triangle configurations in their defining graphs.
• To extend the applicability of Lafforgue’s theorem on the Baum-Connes conjecture to a broad class of Artin groups by verifying the RD condition.
• To provide a combinatorial and analytic proof of RD that relies on geodesic word structure and length-reducing sequences in Artin groups.
• To identify sufficient conditions under which Artin groups satisfy the Baum-Connes conjecture without coefficients, including 3-generation, triangle-freeness, and orientable edge structures.
• To generalize known results on RD and Baum-Connes from braid groups and right-angled Artin groups to a wider class of Artin groups, including those of extra-large type.

Proposed method

• Combines elementary analysis with combinatorial techniques on geodesic words in Artin groups of large type, drawing on prior results from [17] on word structure.
• Uses a length function ℓ on the group to define Sobolev norms ||·||_{2,r,ℓ}, which are compared to the operator norm ||·||_* via the RD inequality: ||φ||_* ≤ C||φ||_{2,r,ℓ}.
• Applies Propositions 7.1 and 7.3 to analyze geodesic factorizations of group elements, particularly focusing on the structure of right and left divisors of elements.
• Employs a case analysis on length-reducing sequences in words, showing that certain configurations (e.g., involving generators a, b, c with m_{ij} ≥ 4) prevent such sequences from passing through middle sections, preserving geodesicity.
• Bounds the number of triples (f₁, Δᵢⱼʳ, f₂) in the set S(g,k,l) by fixing parameters like c^q and e''₁, using polynomial bounds P₁(k) and P²ᵢʲ(k), leading to a finite count for the number of such factorizations.
• Leverages Lafforgue’s result that RD combined with a suitable group action implies the Baum-Connes conjecture without coefficients, applying it to Artin groups satisfying the RD condition.

Experimental results

Research questions

• RQ1 Do Artin groups of large type that exclude triangles with labels (3,3,m) satisfy the rapid decay property?
• RQ2 Under what conditions does the rapid decay property imply the Baum-Connes conjecture without coefficients for Artin groups?
• RQ3 Can the combinatorial structure of geodesic words in Artin groups be used to bound the number of factorizations and verify the RD condition?
• RQ4 Is the rapid decay property satisfied by all Artin groups of extra-large type?
• RQ5 What structural constraints (e.g., 3-generators, triangle-freeness, orientable edges) ensure that an Artin group satisfies the Baum-Connes conjecture without coefficients?

Key findings

• All Artin groups of extra-large type satisfy the rapid decay property, as they satisfy the triangle exclusion condition in Theorem 1.1.
• The rapid decay property holds for all Artin groups of large type whose defining graph has no triangles with edge labels (3,3,m) for finite m ≥ 3.
• For 3-generated Artin groups of large type satisfying the triangle condition, the Baum-Connes conjecture without coefficients holds due to the combination of RD and Lafforgue’s theorem.
• Artin groups with triangle-free defining graphs satisfy the Baum-Connes conjecture without coefficients, as they meet the conditions of Corollary 1.2.
• If the edges of the defining graph can be oriented to avoid two specific subgraph types, the group satisfies the Baum-Connes conjecture without coefficients.
• The number of geodesic factorizations of a group element is bounded by a polynomial in k, leading to a finite count that supports the verification of the RD condition via combinatorial control.