[Paper Review] Finiteness Properties of Chevalley Groups over a Polynomial Ring over a Finite Field

Kai‐Uwe Bux, Ralf Gramlich+1 more|arXiv (Cornell University)|Aug 31, 2009

Coding theory and cryptography1 references7 citations

TL;DR

This paper establishes that Chevalley groups over polynomial rings over finite fields are of type F_{n−1} for any isotropic, absolutely almost simple group over 𝔽_q, without requiring q to be large. The authors generalize previous results by removing the dependence on q's size and extend the finiteness properties to all such groups via geometric and group-theoretic methods involving buildings and arithmetic lattices.

ABSTRACT

It is known from work by H. Abels and P. Abramenko that for a classical Fq-group G of rank n the arithemetic lattice G(Fq[t]) of Fq[t]-rational points is of type Fn-1 provided that q is large enough. We show that the statement is true without any assumption on q and for any isotropic, absolutely almost simple group G defined over Fq.

Motivation & Objective

• To generalize finiteness properties of arithmetic lattices in Chevalley groups over polynomial rings over finite fields.
• To remove the dependence on q being large, which was required in earlier work by Abels and Abramenko.
• To establish that G(𝔽_q[t]) is of type F_{n−1} for any isotropic, absolutely almost simple group G over 𝔽_q.
• To extend the framework to global function fields and S-arithmetic groups, analyzing their finiteness properties via buildings.
• To resolve the negative part of the finiteness property conjecture for S-arithmetic groups using local rank decomposition.

Proposed method

• Utilizes the action of G(𝒪_S) on Euclidean buildings X_p associated with each place p ∈ S.
• Applies geometric group theory techniques, particularly the structure of buildings and their dimensions related to local ranks.
• Employs the theory of arithmetic lattices in algebraic groups over global function fields.
• Leverages results from Bux and Wortman (2007) on the non-finiteness of G(𝒪_S) for F_d with d equal to the sum of local ranks.
• Analyzes the homotopy type and finiteness properties of classifying spaces via group actions on contractible complexes.
• Uses the fact that the dimension of each building X_p equals the local rank of G at p to bound the finiteness type.

Experimental results

Research questions

• RQ1 Is the finiteness property F_{n−1} satisfied by Chevalley groups over 𝔽_q[t] for all isotropic, absolutely almost simple groups, regardless of q's size?
• RQ2 Can the restriction that q must be large, present in earlier results, be removed for G(𝔽_q[t]) to be of type F_{n−1}?
• RQ3 What is the precise finiteness type of S-arithmetic groups G(𝒪_S) in the context of global function fields?
• RQ4 How do the local ranks of G at places in S influence the finiteness properties of G(𝒪_S)?
• RQ5 Does the non-finiteness result for F_d with d equal to the sum of local ranks hold universally for such groups?

Key findings

• The group G(𝔽_q[t]) is of type F_{n−1} for any isotropic, absolutely almost simple algebraic group G over 𝔽_q, without requiring q to be large.
• The result holds uniformly across all finite fields, extending previous results that required q to be sufficiently large.
• The S-arithmetic group G(𝒪_S) is not of type F_d, where d is the sum of the local ranks of G at the places in S, confirming the negative part of the finiteness property conjecture.
• The dimension of the Euclidean building X_p associated with each place p ∈ S equals the local rank of G at p, which is central to the analysis.
• The geometric action of G(𝒪_S) on the product of buildings {X_p}_{p∈S} is used to determine the finiteness properties of the group.
• The paper establishes a complete characterization of the finiteness type of G(𝔽_q[t]) and G(𝒪_S) in terms of the rank and structure of the underlying algebraic group.