[Paper Review] Full residual finiteness growths of nilpotent groups

Khalid Bou-Rabee, Daniel Studenmund|arXiv (Cornell University)|Jun 14, 2014

Geometric and Algebraic Topology6 references4 citations

TL;DR

This paper establishes the full residual finiteness growth of finitely generated nilpotent groups, showing it is polynomial with degree $ c \cdot \dim(G) $ when the center contains the last term of the lower central series. For general nilpotent groups, it introduces a 'terraced filtration' to bound the growth degree, and characterizes when this growth matches word growth—only for virtually abelian groups.

ABSTRACT

Full residual finiteness growth of a finitely generated group $G$ measures how efficiently word metric $n$-balls of $G$ inject into finite quotients of $G$. We initiate a study of this growth over the class of nilpotent groups. When the last term of the lower central series of $G$ has finite index in the center of $G$ we show that the growth is precisely $n^b$, where $b$ is the product of the nilpotency class and dimension of $G$. In the general case, we give a method for finding an upper bound of the form $n^b$ where $b$ is a natural number determined by what we call a terraced filtration of $G$. Finally, we characterize nilpotent groups for which the word growth and full residual finiteness growth coincide.

Motivation & Objective

• To understand the asymptotic behavior of full residual finiteness growth in finitely generated nilpotent groups.
• To determine conditions under which this growth is polynomial and to compute its exact degree.
• To develop a filtration-based method for bounding the growth degree in general nilpotent groups.
• To characterize nilpotent groups for which full residual finiteness growth coincides with word growth.
• To address the challenge that $p$-group quotients vary significantly with prime $p$, complicating global analysis.

Proposed method

• Introduce the concept of a 'terraced filtration'—a chain of normal subgroups $1 = H_0 \leq \cdots \leq H_{c-1} \leq G$ where each $H_i$ satisfies $H_i \cap \gamma_{i+1}(G) = 1$.
• Use distortion results from Osin and Pittet to control word length in quotients and establish detection of balls in finite quotients.
• Construct a generating set $Y$ from basis pulls of successive quotients $H_i/H_{i-1}$ to analyze growth in the Cayley graph.
• Apply a torsion-free power construction $G^{fM}$ to ensure quotients are well-behaved and detectable.
• Bound the index of normal subgroups $N(Dn)$ in $G$ using the dimension of successive layers $H_k/H_{k-1}$, leading to a polynomial upper bound.
• Use the equivalence relation $f \approx g$ to define growth classes independent of generating set, ensuring invariance of the main results.

Experimental results

Research questions

• RQ1 What is the precise polynomial degree of full residual finiteness growth for nilpotent groups where $[Z(G) : \gamma_c(G)] < \infty$?
• RQ2 How can one uniformly bound the full residual finiteness growth of a general finitely generated nilpotent group?
• RQ3 Does the choice of terraced filtration affect the resulting upper bound on growth degree?
• RQ4 For which nilpotent groups does full residual finiteness growth match word growth?
• RQ5 Can the full residual finiteness growth be characterized purely by asymptotic data of the Cayley graph?

Key findings

• When $[Z(G) : \gamma_c(G)] < \infty$, the full residual finiteness growth of $G$ is $\Phi_G(n) \approx n^{c \cdot \dim(G)}$, where $c$ is the nilpotency class.
• For general nilpotent groups, the growth is bounded above by $n^{c \cdot \dim(G) - \sum_{i=1}^{c-1} \dim(H_i)}$ via a terraced filtration $H_0 \leq \cdots \leq H_{c-1}$.
• The upper bound depends on the choice of terraced filtration, as demonstrated by a group $G = U_3 \times U_4 \times U_5 / Z$ with two filtrations yielding different bounds due to $\dim(\pi(U_3)) = 3$ vs. $\dim(\pi(U_4)) = 6$.
• The full residual finiteness growth $\Phi_G$ coincides with word growth $w_G$ if and only if $G$ is virtually abelian.
• The growth function $\Phi_G$ is independent of the choice of finite generating set, as established via the equivalence $f \approx g$.
• The construction of $G/N(Dn)$ detects the ball $B_Y(n)$, and its order is bounded by a polynomial of degree $c \cdot \dim(G) - \sum_{i=1}^{c-1} \dim(H_i)$, proving the upper bound.