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$S$ -arithmetic groups acting simply transitively on products of Bruhat-Tits trees

This paper affirmatively answers Lubotzky's question by constructing $S$ -arithmetic lattices derived from totally definite quaternion algebras that act simply transitively on Bruhat-Tits trees and their products, thereby generating new examples of Ramanujan Cayley graphs and regular cubical complexes.

Original authors: Jonah Mendel, Jiahui Yu

Published 2026-06-09

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CC BY 4.0

Original authors: Jonah Mendel, Jiahui Yu

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 architect trying to build a perfect, infinite city. This city isn't made of bricks, but of mathematical "trees" (branching structures that go on forever in every direction). Your goal is to create a group of "builders" (mathematical symmetries) that can walk through this city and visit every single intersection exactly once without ever stepping on the same spot twice or skipping a spot.

In the world of mathematics, this is called a simply transitive action. It's the "Goldilocks" zone of movement: not too crowded (where you step on the same spot twice), not too sparse (where you miss spots), but just right.

For decades, mathematicians have known how to build these cities and how to get builders who visit most spots, but finding builders who hit every single spot exactly once has been a massive puzzle. This paper, by Jonah Mendel and Jiahui Yu, solves that puzzle for a specific, very important type of city.

Here is a breakdown of their discovery using simple analogies:

1. The City and the Trees

Think of the "city" as a product of several Bruhat–Tits trees.

The Trees: Imagine a tree where every branch splits into $N$ new branches. In math, these represent different "directions" or "layers" of the city.
The City: Sometimes, the city is just one tree. Sometimes, it's two trees woven together (like a grid), or even more.
The Goal: The authors want to find a specific set of rules (a "group") that allows a traveler to start at a point, follow the rules, and land on every single intersection in the city exactly once.

2. The Old Problem: The "Almost" Builders

Before this paper, mathematicians (like Lubotzky) knew how to make builders who could visit the city, but they often had to visit some spots multiple times or skip others. They asked: "Can we refine these rules so that every spot is visited exactly once?"

For a long time, the answer was "We don't know," or "Only in very specific, simple cases."

3. The New Solution: The "Complementary Triple"

The authors' main invention is a tool they call a "Complementary Triple."

Think of this like a lock and key system combined with a traffic cop:

The Lock (The Group): This is the set of rules the builders must follow.
The Key (The Subgroup): This is a specific subset of rules that fits perfectly into the lock.
The Traffic Cop (The Complement): This ensures that the Key doesn't overlap with the Lock in a way that causes traffic jams (revisiting spots).

The authors developed a method to calculate these "Keys" and "Traffic Cops" using computer programs (specifically a tool called Magma). They proved that for a huge class of mathematical cities (those built from "totally definite quaternion algebras" with "class number one"), you can almost always find a perfect Key and Traffic Cop.

The Result: For almost any prime number you choose, they can construct a group of builders that walks through the city, visiting every single intersection exactly once.

4. Why Does This Matter? (The "Ramanujan" Connection)

The paper mentions that these perfect walking patterns create Ramanujan graphs and Ramanujan complexes.

The Analogy: Imagine you are designing a communication network (like the internet or a phone system). You want the network to be incredibly efficient. You want information to travel from any point to any other point as quickly as possible, without getting stuck in loops or bottlenecks.
The "Ramanujan" Property: In math, a "Ramanujan" graph is the most efficient network possible. It has the best possible "traffic flow" for its size.
The Paper's Contribution: By finding these "simply transitive" builders, the authors have created a factory for building these super-efficient networks. They aren't just one-off examples; they can build infinite families of them.

5. The "Higher Dimensions" Twist

Most previous work only looked at cities made of one tree (1D) or two trees (2D).

The Breakthrough: This paper shows how to do this for cities made of many trees woven together (higher dimensions).
The Analogy: Imagine moving from a flat map (2D) to a 3D skyscraper, or even a 4D hyper-structure. The authors figured out how to build the perfect "elevator system" (the group) that visits every floor and every room in these complex, multi-layered structures exactly once.

6. The "No Clumps" Guarantee (Torsion-Free)

In math, sometimes a builder might get stuck in a loop, returning to the start after a few steps (this is called "torsion").

The authors also figured out how to ensure their builders never get stuck in loops. They developed a "torsion obstruction" test (a checklist) to make sure the builders keep moving forward forever without repeating a cycle. This is crucial for creating "clean" mathematical structures.

Summary

Jonah Mendel and Jiahui Yu solved a decades-old puzzle about how to perfectly navigate complex mathematical cities.

They invented a new tool (Complementary Triples) to find the perfect navigation rules.
They proved these rules exist for a vast array of mathematical structures.
They extended this from simple 1D/2D structures to complex, multi-dimensional structures.
They ensured these rules create the most efficient networks possible (Ramanujan graphs) without any "traffic jams" (loops).

This work provides a new, infinite supply of "perfect" mathematical blueprints that can be used to build highly efficient networks and solve deep problems in number theory.

Technical Summary: S-Arithmetic Groups Acting Simply Transitively on Products of Bruhat–Tits Trees

Problem Statement
The paper addresses a question posed by Lubotzky regarding the existence of simply transitive actions by $p$ -arithmetic groups on the vertices of Bruhat–Tits trees. While Margulis and Lubotzky–Phillips–Sarnak (LPS) established that certain arithmetic groups act transitively on these trees, and that principal congruence subgroups often act simply transitively, a general method to construct such simply transitive subgroups for maximal orders of class number one in totally definite quaternion algebras over arbitrary number fields remained open.

The problem extends to higher dimensions: constructing $S$ -arithmetic lattices that act simply transitively on products of Bruhat–Tits trees (specifically, products of trees of the same valency). Such actions are crucial for generating Ramanujan Cayley graphs and Ramanujan Cayley regular cubical complexes, which have applications in coding theory and the study of lattices acting on products of trees (Burger–Mozes groups).

Methodology
The authors employ a combination of arithmetic classification, finite group theory, and computational algebra to resolve the problem.

Classification of Orders: The work relies on the classification of totally definite quaternion algebras with maximal orders of class number one, as enumerated by Kirschmer and Voight. This provides a finite list of algebras over number fields of degree at most 5.
Complementary Triples: The core technical innovation is the introduction of complementary triples $(T, \pi, H)$ $(T, π, H)$ .
- $T$ is a set of ramified primes.
- $\pi$ is an injective homomorphism from the $T$ -arithmetic group $\Gamma_{O,T}$ into a finite group $G$ (typically a reduction modulo an ideal $C$ ).
- $H$ is a subgroup of $G$ complementary to the image $\pi(\Gamma_{O,T})$ (i.e., $H \cap \pi(\Gamma_{O,T}) = \{1\}$ and $H \cdot \pi(\Gamma_{O,T}) = G$ ).
- The existence of such a triple allows the construction of a subgroup $\Gamma = \pi^{-1}(H)$ which acts simply transitively on the vertices of the product of trees.
Finite Group Computations: The authors utilize the computer algebra system Magma to:
- Enumerate units and normalizers for the specific orders in the Kirschmer–Voight classification.
- Compute permutation representations of the finite quotients $G[C]$ .
- Search for complementary subgroups $H$ within these finite groups.
- Verify torsion-freeness using torsion obstruction sets derived from trace-zero elements and reduced norms.
Metacommutation: The paper utilizes the theory of metacommutation (Chari) to derive explicit presentations for the resulting lattices, ensuring they possess the necessary Cayley structure.

Key Contributions and Results

Theorem A (Single Tree Case): For any totally real number field $K$ , totally definite quaternion algebra $B/K$ , and maximal order $O$ of class number one, there exists a congruence arithmetic lattice $\Gamma$ commensurable with $\Gamma_{O,p}$ that acts simply transitively on the vertices of the Bruhat–Tits tree $T(K_p)$ for all but finitely many prime ideals $p$ . This confirms Lubotzky's question affirmatively for this class of orders.
Theorem B (Product of Trees): The result is extended to $S$ -arithmetic settings. For any finite set of primes $S$ (disjoint from a fixed finite set), there exists a congruence lattice acting simply transitively on the product of trees $T_{S_0} = \prod_{p \in S_0} T(K_p)$ .
Theorem C (Same Valency): The authors construct infinitely many $S$ -arithmetic lattices derived from quaternion algebras over number fields that act simply transitively on a product of $n$ trees of the same valency for $1 \leq n \leq 5$ . This provides the first explicit examples of irreducible arithmetic groups with this property defined over number fields (previously known only in positive characteristic by Stix and Vdovina).
Torsion-Freeness: The paper establishes criteria (Proposition 43) to ensure the constructed lattices are torsion-free by analyzing the intersection of the image of the group with the set of trace-zero reduced norms in finite quotients. Theorem 50 proves that for definite class number one algebras over $\mathbb{Q}$ , there are infinitely many torsion-free $S$ -arithmetic groups arising from these constructions.
Explicit Presentations: The paper provides explicit group presentations for these lattices (e.g., Examples 18 and 19), demonstrating how metacommutation relations define the group structure.

Significance
The paper claims significance in several areas:

Resolution of an Open Problem: It answers Lubotzky's question regarding the existence of simply transitive actions for maximal orders of class number one, a problem that had remained largely open.
New Examples of Ramanujan Objects: The constructions yield infinite families of new Ramanujan Cayley graphs and Ramanujan Cayley regular cubical complexes. These are significant in spectral graph theory and the study of high-dimensional expanders.
Arithmetic Lattices on Products of Trees: By providing examples over number fields where the tree factors have the same valency, the work fills a gap in the theory of lattices acting on products of trees (Burger–Mozes groups), which previously relied heavily on function field analogues or specific cases over $\mathbb{Q}$ with differing valencies.
Computational Classification: The paper provides a complete computational classification of minimal complementary triples for the 49 totally definite quaternion algebras of class number one identified by Kirschmer and Voight, serving as a foundational resource for constructing such groups.
Connections to Other Fields: The authors note the relevance of these constructions to non-archimedean uniformization of complex surfaces, the study of Ramanujan subshifts, and the development of lossless vertex expanders and potential quantum locally testable codes.

The work is presented as a rigorous arithmetic construction, relying on the classification of orders and finite group computations, without proposing new experimental protocols or unverified applications beyond the established connections to coding theory and expanders mentioned in the introduction.