[Paper Review] Subgroups of the upper-triangular matrix group with maximal derived length and a minimal number of generators

S. P. Glasby|arXiv (Cornell University)|Oct 19, 2014

graph theory and CDMA systems3 references3 citations

TL;DR

This paper constructs explicit 2- and 3-generated subgroups of the upper-triangular unipotent matrix group $U_n(bF)$ over a field $bF$ that achieve the maximal possible derived length $d = \lceil \log_2 n \rceil$. It proves that such a 2-generated subgroup exists if and only if $\frac{21}{32}2^d < n \leq 2^d$, using recursive commutator constructions based on binary tree structures and polynomial identities in free groups to ensure nontriviality at the derived length $d$. The results are constructive, with explicit generating matrices provided via recursive formulas.

ABSTRACT

The group U_n(F) of all nxn unipotent upper-triangular matrices over F has derived length d := Ceiling(log_2 (n)), equivalently 2^{d-1} &lt; n &lt;= 2^d. We prove that U_n(F) has a 3-generated subgroup of derived length d, and it has a 2-generated subgroup of derived length d if and only if (21/32)* 2^d &lt; n &lt;= 2^d.

Motivation & Objective

• To determine the minimal number of generators required for subgroups of $U_n(\bbF)$ to achieve the maximal derived length $d = \lceil \log_2 n \rceil$.
• To characterize precisely for which $n$ the group $U_n(\bbF)$ admits a 2-generated subgroup of derived length $d$.
• To provide explicit, recursive constructions of such generating matrices using commutator identities and free group word constructions.
• To establish the asymptotic density of $n$ for which $U_n(\bbF)$ has a 2-generated subgroup of maximal derived length, showing $\liminf \pi(N) = \frac{11}{21}$ and $\limsup \pi(N) = \frac{11}{16}$.

Proposed method

• Uses recursive lifting via a surjective homomorphism $\pi: U_{2n-1} \to U_n \times U_n$ to construct 3-generated subgroups from smaller ones.
• Employs a binary tree structure with $d$ layers, where each layer corresponds to elements in the derived series, and uses commutators of matrices at layer $k$ to generate elements at layer $k+1$.
• Applies a cyclic re-use of generators across layers to reduce the number of required generators from $n-1$ to 3, by defining words in the free group $F = \langle x_1,x_2,x_3 \rangle$ of rank 3.
• For 2-generated subgroups, constructs words in the free group of rank 2 using nested commutators and polynomial identities, ensuring that the resulting matrix entry has coefficient $\pm 1$ via the Multiplication Lemma.
• Verifies nontriviality of the top-level commutator by analyzing monomial summands in the group algebra, ensuring that no summand divides another across different generator types.
• Uses induction on $d$, with base cases for $d=1$ to $d=5$, and proves that $F^{(d-1)} \subseteq \gamma_{21 \cdot 2^{d-5}}(F)$ for $d \geq 5$, linking group word length to matrix size constraints.

Experimental results

Research questions

• RQ1 For which $n$ does $U_n(\bbF)$ admit a 2-generated subgroup of derived length $d = \lceil \log_2 n \rceil$?
• RQ2 What is the minimal number of generators required for a subgroup of $U_n(\bbF)$ to achieve maximal derived length?
• RQ3 Can such subgroups be constructed explicitly using recursive formulas based on commutator identities?
• RQ4 What is the asymptotic density of $n$ for which $U_n(\bbF)$ has a 2-generated subgroup of maximal derived length?
• RQ5 How does the structure of the derived series relate to the combinatorics of monomial summands in free group words?

Key findings

• The group $U_n(\bbF)$ has a 3-generated subgroup of derived length $d = \lceil \log_2 n \rceil$ for all $n$ satisfying $2^{d-1} < n \leq 2^d$.
• A 2-generated subgroup of derived length $d$ exists in $U_n(\bbF)$ if and only if $\frac{21}{32}2^d < n \leq 2^d$.
• The proportion $\pi(N)$ of $n \leq N$ for which $U_n(\bbF)$ has a 2-generated subgroup of maximal derived length satisfies $\frac{11}{21} < \pi(N) \leq 1$, with $\liminf \pi(N) = \frac{11}{21}$ and $\limsup \pi(N) = \frac{11}{16}$.
• For $d \geq 5$, the derived length $d$ subgroup exists only when $n > 21 \cdot 2^{d-5}$, due to the containment $F^{(d-1)} \subseteq \gamma_{21 \cdot 2^{d-5}}(F)$ in the free group.
• The construction is recursive and explicit: generating matrices are defined via lifting under $\pi: U_{2n-1} \to U_n \times U_n$, with words in the free group defined recursively.
• Nontriviality of the top-level commutator is verified by showing that a monomial summand in the matrix entry has coefficient $\pm 1$, using the Multiplication Lemma and congruence conditions on the number of $\alpha$'s in monomials.