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Surjectivity of Engel Words on $\mathrm{SL}_2(\mathcal{O})$ and $\mathrm{PSL}_2(\mathcal{O}_2)$

This paper establishes that Engel word maps are surjective on $\mathrm{PSL}_2(\mathcal{O}_2)$ and cover all lifts of non-scalar elements in $\mathrm{SL}_2(\mathcal{O})$ for sufficiently large residue fields, thereby generalizing previous results on fields to local principal ideal rings.

Original authors: Ayon Roy, Anupam Singh

Published 2026-06-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ayon Roy, Anupam Singh

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master locksmith trying to open a very specific, complex safe. The safe is a mathematical structure called a group, and the "key" you are trying to find is a specific combination of numbers (matrices) that, when mixed together in a certain way, can produce any possible combination inside the safe.

This paper, written by Ayon Roy and Anupam Singh, is about testing whether a specific type of "key-making machine" (called an Engel word map) can produce every single possible combination in two types of safes: SL2(O) and PSL2(O2).

Here is the breakdown of their adventure using simple analogies:

1. The Machine: The "Engel Word"

Think of the Engel word as a recipe for mixing ingredients.

You start with two ingredients, let's call them X and Y.
The recipe says: "Mix X and Y, then take that result and mix it with Y again, then mix that result with Y again..."
If you do this m times, you get a final product.
The big question the authors ask is: If I give you any random target combination (a matrix) inside the safe, can I always find two starting ingredients (X and Y) that, when run through this recipe, produce exactly that target?

2. The Safes: Local Rings (The "Layers")

The authors aren't just looking at simple safes (like those over a standard field of numbers). They are looking at safes built with local rings.

The Analogy: Imagine a safe made of layers of clay.
- The bottom layer is a simple, flat field (like a pond of water).
- The top layers are "lifts" or thicker layers of clay sitting on top of that water.
- SL2(O) is a very tall tower of these clay layers (a "complete" ring).
- PSL2(O2) is a shorter tower, specifically with just two layers (the bottom water and one layer of clay on top).

The challenge is harder here because the "clay" (the ring) has special properties that make mixing ingredients slightly different than in the water below.

3. The Main Discovery: "Lifting" the Solution

The authors prove a powerful concept called Surjectivity (or "completeness"). They show that if you can make a specific combination in the bottom layer (the water/field), you can almost always "lift" that solution up to the higher clay layers.

The "Non-Scalar" Rule: They found that for almost any target combination (as long as it's not a boring, uniform "scalar" type), if you have enough "space" in your numbers (a large enough field size, denoted as q), you can always find the starting ingredients X and Y to make it work, even in the thick clay layers.
The "Scalar" Exception: There are a few special, uniform combinations (like the identity matrix I or -I) that are tricky. The authors had to use special tricks (like checking if -1 can be written as a sum of two squares) to prove that even these special cases can be made.

4. The Two Big Results

The paper delivers two main "wins":

The Tall Tower (SL2(O)): If the bottom layer of numbers is big enough (specifically, if the number of elements q is larger than a certain threshold, roughly $2 \cdot 3^{2m+2}$ ), then every single non-boring combination in the tall tower can be created by the Engel machine. You just need to find the right starting ingredients.
The Short Tower (PSL2(O2)): For the shorter tower with exactly two layers, they proved that the Engel machine is completely surjective. This means it can create every single possible combination (including the tricky uniform ones) in this specific two-layer structure, provided the bottom layer is large enough.

5. How They Did It (The Toolkit)

To solve this, the authors used a few clever tools:

The Trace Map (The "Fingerprint"): Instead of trying to match the whole complex matrix, they looked at a single number called the "trace" (the sum of the diagonal numbers). They proved that if you can match the "fingerprint" (trace) of the target, you can usually reconstruct the whole matrix.
Hensel's Lemma (The "Ladder"): This is a mathematical technique that allows you to take a solution found in the bottom layer and "climb the ladder" step-by-step to the higher layers, fixing small errors as you go up.
Magnus Embedding: They used a special mathematical "embedding" (like putting a 2D drawing into a 3D model) to handle the tricky "unipotent" elements (matrices that look like they are sliding rather than rotating).

Summary

In simple terms, Roy and Singh showed that for these specific mathematical structures, the "Engel mixing machine" is incredibly powerful. As long as the underlying number system is large enough, this machine can generate every possible outcome in the system, whether you are working in a simple two-layer structure or a complex, infinite tower of layers. They solved the puzzle of "lifting" solutions from the simple world to the complex world, confirming that the machine works perfectly under the right conditions.

Technical Summary: Surjectivity of Engel Words on $SL_2(\mathcal{O})$ and $PSL_2(\mathcal{O}_2)$

Problem Statement
This paper investigates the surjectivity of Engel word maps on special linear groups over local principal ideal rings. Specifically, it addresses whether every element in $SL_2(\mathcal{O})$ and $PSL_2(\mathcal{O}_2)$ can be expressed as an $m$ -th Engel word $e_m(x,y) = [\dots[[x,y],y],\dots,y]$ . The study focuses on two primary settings:

$SL_2(\mathcal{O})$ , where $\mathcal{O}$ is a complete local principal ideal ring with residue field $k \cong \mathbb{F}_q$ of characteristic $\neq 2$ .
$PSL_2(\mathcal{O}_2)$ , where $\mathcal{O}_2$ is a local principal ideal ring of length 2 (i.e., $\mathcal{O}/\pi^2\mathcal{O}$ ).

The work extends known results from fields (specifically $\mathbb{F}_q$ ) to these ring structures, addressing a lifting problem: if an element $\bar{A} \in SL_2(k)$ is in the image of the Engel map, does every lift $A \in SL_2(\mathcal{O})$ also lie in the image?

Methodology
The authors employ a multi-faceted approach combining group theory, Lie algebra techniques, and polynomial lifting strategies:

Lifting Solutions via Hensel's Lemma: A core technique involves lifting solutions of multivariable polynomial equations from the residue field $k$ to the ring $\mathcal{O}_2$ and subsequently to $\mathcal{O}$ . The authors establish conditions under which simple roots of trace polynomials (derived from Engel words) can be lifted. This relies on the Jacobian criterion to ensure the existence of lifts for regular semisimple elements.
Trace Polynomials and Jacobian Analysis: The paper derives explicit trace polynomials $\Psi_m(s,u,v)$ corresponding to Engel words, where $s, u, v$ are traces of $X, XY, Y$. The authors analyze the Jacobian of these polynomials to determine when the derivative map of the Engel word is surjective at the Lie algebra level ( $\mathfrak{sl}_2(k)$ ).
Coadjoint Action and Fixed Vectors: To handle the lifting of non-scalar elements, the authors utilize the coadjoint action of $SL_2(k)$ on the dual Lie algebra $\mathfrak{sl}_2^*(k)$ . They prove that if the subgroup generated by the pre-images of an element has no non-trivial fixed vectors under this action, the lifting is possible. This generalizes methods used by Avni, Gelander, Kassabov, and Shalev for commutators.
Magnus Embedding: For unipotent elements, the authors utilize a generalization of the Magnus embedding into a group of upper triangular matrices over Laurent polynomials to demonstrate that specific unipotent matrices are images of Engel words.
Conjugacy Class Analysis: The paper provides a detailed classification of $GL$-conjugacy classes in $SL_2(\mathcal{O}_2)$ , distinguishing between regular semisimple, scalar, and non-regular semisimple (unipotent) elements to treat them separately.

Key Contributions and Results

Lifting of Non-Scalar Elements (Theorem 10.1):
The authors prove that for a complete local principal ideal ring $\mathcal{O}$ with residue field $\mathbb{F}_q$ (characteristic $\neq 2$ ), there exists a constant $q_0(m) = 2 \cdot 3^{2m+2}$ such that for all $q \geq q_0(m)$ , every lift of a non-scalar element $A \in SL_2(k)$ that is in the image of the $(m+1)$ -th Engel map over $k$ is also in the image of the $(m+1)$ -th Engel map over $SL_2(\mathcal{O})$ . This result covers regular semisimple elements and specific unipotent elements.
Surjectivity on $PSL_2(\mathcal{O}_2)$ (Theorem 10.4):
The paper establishes that for a local principal ideal ring $\mathcal{O}_2$ of length 2 with residue field $\mathbb{F}_q$ ( $q \geq q_0(m)$ ), the $(m+1)$ -th Engel word map is surjective on $PSL_2(\mathcal{O}_2)$ . This generalizes previous results by Bandman et al. for fields.
Handling Scalar Lifts:
- Identity Lifts: The authors prove that any lift of the identity matrix $I$ in $SL_2(\mathcal{O}_2)$ is a commutator (and thus an Engel word image) provided $|k| \geq 5$ .
- Negative Identity Lifts: They show that lifts of $-I$ are commutators if $-1$ is a sum of two squares in the residue field (which is always true for finite fields of odd characteristic).
- Unipotent Lifts: Using the Magnus embedding technique, they demonstrate that lifts of unipotent elements (e.g., $\begin{pmatrix} 1 & \pi \\ 0 & 1 \end{pmatrix}$ ) are in the image of Engel maps for sufficiently large $q$ .
Technical Lemmas on Trace Maps:
The paper provides recursive formulas for the trace of Engel words and proves that the derivative map of the Engel word is surjective under specific conditions regarding the centralizers of the generating elements in the Lie algebra.

Significance and Claims
The paper claims to generalize existing results on the surjectivity of Engel words from fields to local rings, specifically addressing the "lifting problem" for linear groups over local principal ideal rings. The authors position their work as a step toward resolving Shalev's conjecture regarding the surjectivity of commutators and Engel words in $SL_n(\mathbb{Z}_p)$ and similar structures.

The significance lies in:

Extending the "Waring-like" problem (expressing group elements as products of word map images) to the context of local rings.
Providing a constructive method for lifting solutions from the residue field to the ring level using trace polynomials and coadjoint actions.
Resolving the surjectivity for $PSL_2(\mathcal{O}_2)$ , a case that was previously open for higher Engel words.

The authors explicitly state that their results rely on the residue field size $q$ being sufficiently large ( $q \geq 2 \cdot 3^{2m+2}$ ) and the characteristic not being 2. They do not claim surjectivity for all $q$ or for $SL_2(\mathcal{O})$ in the projective case without the length-2 restriction, noting that the problem remains open for general fields and rings in other contexts.