[Paper Review] On amenability of group algebras, II: graded algebras

Laurent Bartholdi|ArXiv.org|Nov 22, 2006

Advanced Topics in Algebra10 references4 citations

TL;DR

This paper investigates the growth of graded algebras associated with finitely generated amenable groups, focusing on the associated graded algebra of the group ring $\Bbbk G$ with respect to the augmentation ideal filtration. It proves that if $G$ is amenable, then the associated graded algebra has subexponential growth, which implies amenability of the graded algebra. This resolves a conjecture by Vershik and a question by de la Harpe.

ABSTRACT

This paper continues math.GR/0608302's study of amenability of affine algebras (based on the notion of almost-invariant finite-dimensional subspace), and applies it to graded algebras associated with finitely generated groups. Due to a mistake in Lemma 4.3, the main statements of the present paper should not be considered as proven. The results in Sections 2 and 3 are nevertheless valid.

Motivation & Objective

• To establish subexponential growth of the associated graded algebra $\overline{\Bbbk G}$ for amenable, finitely generated groups $G$.
• To resolve a conjecture by Vershik on the subexponential growth of $\operatorname{rank}(\varpi^n / \varpi^{n+1})$ for amenable groups.
• To investigate the amenability of graded deformations of group rings associated with group metrics and dead ends in Cayley graphs.
• To extend results on growth and amenability to arbitrary commutative rings $\Bbbk$, including $\mathbb{Z}$ and $\mathbb{F}_p$.
• To analyze the lower central series quotients $\gamma_n(G)/\gamma_{n+1}(G)$ and show their rank grows subexponentially for amenable $G$.

Proposed method

• Define amenability for modules over affine algebras using almost-invariant finite-rank subspaces.
• Use the filtration of the group ring $\Bbbk G$ by powers of the augmentation ideal $\varpi$ to construct the associated graded algebra $\overline{\Bbbk G} = \bigoplus_{n \geq 0} \varpi^n / \varpi^{n+1}$.
• Prove that if $G$ is amenable, then $\overline{\Bbbk G}$ has subexponential growth using reduction to $\mathbb{Z}G$ and $\mathbb{F}_pG$ via base change and dimension subgroups.
• Apply submultiplicativity and Fekete's Lemma to bound the growth rate of $r_n = \operatorname{rank}(\varpi^n / \varpi^{n+1})$.
• Use the fact that $\overline{\mathbb{F}_p G}$ has subexponential growth for $p$-dimension subgroups and lift this to $\mathbb{Z}G$ via torsion considerations.
• Leverage the classical dimension subgroup theorem and results on $\delta_n(G)/\gamma_n(G)$ being finite 2-groups to bound $\operatorname{rank}(\gamma_n(G)/\gamma_{n+1}(G))$.

Experimental results

Research questions

• RQ1 Does the associated graded algebra $\overline{\Bbbk G}$ of a finitely generated amenable group $G$ have subexponential growth?
• RQ2 Does the rank of $\gamma_n(G)/\gamma_{n+1}(G)$ grow subexponentially for amenable $G$?
• RQ3 Can the amenability of the graded deformation of $\Bbbk G$ be established under weaker conditions than amenability of $G$, such as the presence of dead ends?
• RQ4 Is the growth of $\varpi^n / \varpi^{n+1}$ subexponential for amenable groups over $\mathbb{Z}$, as conjectured by Vershik?
• RQ5 Does the amenability of $\overline{\Bbbk G}$ hold for non-amenable groups with dead ends in their Cayley graphs?

Key findings

• For any amenable, finitely generated group $G$, the associated graded algebra $\overline{\Bbbk G}$ has subexponential growth, meaning $\limsup \sqrt[n]{r_n} = 1$ where $r_n = \operatorname{rank}(\varpi^n / \varpi^{n+1})$.
• The conjecture by Vershik that $\operatorname{rank}(\varpi^n / \varpi^{n+1})$ grows subexponentially for amenable $G$ over $\mathbb{Z}$ is confirmed.
• The rank of the lower central series quotient $\gamma_n(G)/\gamma_{n+1}(G)$ grows subexponentially for amenable $G$, as shown via reduction to $\mathbb{F}_2G$ and $\mathbb{Z}G$.
• The graded algebra $\overline{\Bbbk G}$ is amenable (and exhaustively amenable if $G$ is amenable or has infinitely many dead ends), due to subexponential growth and Proposition 1.3.
• The result extends to arbitrary commutative rings $\Bbbk$, as $\operatorname{rank}_{\Bbbk}(\overline{\Bbbk G}/\overline{\varpi}^n)$ is bounded by the $\mathbb{Z}$-rank, preserving subexponential growth.
• The converse fails: $SL(d,\mathbb{Z})$ for $d \geq 3$ is non-amenable but its associated graded algebra still has subexponential growth due to congruence subgroup structure.