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Non-residually finite $\tilde{C}_2$ -lattices

This paper presents the first known examples of non-residually finite lattices on irreducible buildings, which include the first simple CAT(0)-groups with property (T) and the first CAT(0)-groups not quasi-isometric to a direct product, while also classifying type-preserving vertex-regular lattices on $\tilde{A}_2$ buildings of thickness three and identifying a new arithmetic example.

Original authors: Thomas Titz Mite, Stefan Witzel

Published 2026-08-21

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Thomas Titz Mite, Stefan Witzel

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

In the vast landscape of modern mathematics, there is a field dedicated to understanding symmetry and shape through the lens of groups. A group, in this context, is simply a collection of symmetries that can be combined and reversed, much like the ways one can rotate or flip a geometric object. For over a century, mathematicians have been particularly interested in "lattices," which are specific types of groups that act on complex geometric structures called buildings. These buildings are not made of brick and mortar but are intricate, high-dimensional networks of triangles and squares that stretch out infinitely in every direction. A key question that has long puzzled researchers is whether these lattices are "residually finite." This property essentially asks if a group is made up of enough smaller, finite pieces to be fully understood by looking at its finite shadows. If a group is residually finite, it means that for any distinct move within the group, there is a finite map where that move does not look like doing nothing. If it is not, the group contains hidden, infinite complexities that cannot be detected by any finite test. For decades, the known examples of lattices on these exotic, non-standard buildings were suspected to be non-residually finite, but no one could prove it for a single case.

A team of researchers has now provided the first concrete proof that such groups exist. They constructed five specific, finite shapes made of triangles that serve as the blueprint for these infinite structures. When these shapes are unfolded into their infinite versions, they create what are known as exotic buildings of a type called $\tilde{C}_2$ . The fundamental groups associated with these shapes—the mathematical descriptions of how one can walk around loops within them—have been shown to be non-residually finite. This means that within these groups, there are specific, non-trivial moves that look like doing nothing in every possible finite version of the group. The researchers did not just guess this; they used a combination of computer-assisted searches to find the right shapes and rigorous mathematical verification to confirm that the resulting groups possess this elusive property.

The discovery is significant because it breaks a long-standing barrier in the field. Before this work, the only known examples of non-residually finite lattices existed on structures that were essentially products of trees, which are simpler, one-dimensional networks. The new examples are "irreducible," meaning they cannot be broken down into simpler products; they are genuinely two-dimensional and complex. This finding also reveals that the finite residuals of these groups are "simple," a term meaning they have no non-trivial normal subgroups to hide behind, making them structurally very rigid. Furthermore, these groups possess a property called "Kazhdan's property (T)," which implies a kind of rigidity where the group resists being deformed or approximated by simpler structures. This combination of having a simple finite residual, being rigid, and being non-residually finite was previously unknown in the world of lattices on irreducible buildings.

To find these examples, the authors employed a massive computational search. They explored a vast space of possible triangle complexes, looking for those that satisfied specific geometric rules ensuring the resulting infinite structure was a valid building. They found five such complexes, labeled with indices to distinguish them. For the first example, involving a thickness of three, the group is so rigid that it is its own finite residual, meaning the entire group is the hidden, infinite part that cannot be seen in finite quotients. For the other four examples, involving a thickness of four, the situation is slightly different but still confirms the non-residually finite nature. The researchers verified these results by checking that certain complex loops within the groups could not be reduced to nothing in any finite setting, a task that required extensive computer calculations to handle the sheer number of possibilities.

Beyond the main discovery, the paper also offers a complete classification of a different, well-behaved type of lattice on a related but distinct type of building. The researchers cataloged all possible lattices that act regularly on the vertices of a building of type $\tilde{A}_2$ with a specific thickness of three. They found exactly thirteen such lattices. Most of these were already known to be "arithmetic," meaning they arise from number theory and algebraic equations. However, one of the thirteen turned out to be a new, previously unknown arithmetic example. This specific lattice is built using a field of numbers involving the square root of negative twenty-three, a detail that connects the geometric structure to deep properties of number theory. The other twelve lattices in this classification act on "exotic" buildings, which are not the standard ones derived from number theory, further highlighting the diversity of these geometric structures.

The implications of these findings extend to how mathematicians view the relationship between geometry and algebra. The paper demonstrates that these new lattices are not just different from the old ones; they are fundamentally distinct in their large-scale geometry. Using a concept called quasi-isometry, which measures how shapes look when viewed from a great distance, the authors proved that none of the new lattices can be transformed into one another or into any previously known lattice without distorting their essential structure. This means that the mathematical universe of these groups is much richer and more varied than previously thought. The work also provides a new method for determining the full symmetry group of these buildings, showing that for the new examples, the group of symmetries is discrete and finite in its extensions, a property that helps distinguish them from other known structures.

The researchers used a clever strategy to prove their main result, relying on the fact that these new lattices contain subgroups that are already known to be non-residually finite. By embedding these known "bad" subgroups into the new, larger structures, they ensured that the larger groups inherited the same hidden complexities. They then used a computer to verify that the new groups did not accidentally acquire extra symmetries that would make them behave differently. This process involved reconstructing finite "balls" of the infinite structure and checking their local symmetries, a task that confirmed the groups were as rigid and unique as the theory predicted. The result is a set of five new, concrete examples that stand as the first verified instances of non-residually finite lattices on irreducible buildings, opening the door for further exploration into the hidden depths of geometric symmetry.

Technical Summary: Non-residually finite $\tilde{C}_2$ -lattices

Problem Statement

The paper addresses a longstanding open problem in geometric group theory and the theory of buildings: the existence of non-residually finite lattices on irreducible Euclidean buildings. While it is well-established that lattices on products of trees (reducible buildings) can be non-residually finite (e.g., Burger–Mozes–Wise groups), and while it is conjectured that lattices on irreducible "exotic" buildings (those not arising from algebraic groups) are generally not residually finite, no explicit example had been verified until this work. Specifically, the authors aim to construct and verify the first examples of non-residually finite lattices on irreducible $\tilde{C}_2$ -buildings.

Methodology

The authors employ a combination of computational search, combinatorial group theory, and geometric rigidity arguments.

Computational Search via Radu Graphs:
The core of the construction relies on a search algorithm for finite non-positively curved combinatorial chamber complexes (GABs). The authors utilize the concept of "Radu graphs," which encode the local geometry of the complex. They define a "score" function based on the size of a partial triangle cover and perform a greedy search through the orbit of Radu graphs under type-preserving permutations. This search identified specific complexes $Y^q_i$ that serve as fundamental domains for the desired lattices.
Embedding Non-Residually Finite Subgroups:
To prove non-residual finiteness, the authors construct the new lattices $\Gamma^q_i$ such that they contain known non-residually finite subgroups as subcomplexes.
- For the case $q=2$ , the complex $Y^2_1$ contains a subdivision of the square complex $S_R$ associated with the Radu group $\Gamma_R$ (a BMW-group on a product of trees).
- For the case $q=3$ , the complexes $Y^3_k$ contain a subdivision of the square complex $S_{JW}$ associated with the Janzen–Wise group $\Gamma_{JW}$ .
  By Lemma 2.2 (injectivity of fundamental groups of locally convex subcomplexes), the fundamental groups of these subcomplexes embed into $\Gamma^q_i$ . Since the subgroups are known to be non-residually finite, the larger groups inherit this property.
Normal Forms and Rigidity:
To analyze the groups further (specifically to determine automorphism groups and verify simplicity), the authors develop an algorithm for computing "normal forms" for words in these lattices. This involves defining canonical paths in the universal cover (buildings) and establishing rewriting rules. This algorithm allows for the computation of finite balls in the Cayley graphs, which are used to prove that the lattices are the full automorphism groups of their respective buildings (or have specific finite index extensions).
Classification of $\tilde{A}_2$ -lattices:
As a secondary but independent result, the authors perform an exhaustive search to classify all type-preserving, vertex-regular lattices on $\tilde{A}_2$ -buildings of thickness 3. This involves enumerating all possible GABs with specific combinatorial constraints and analyzing their fundamental groups.

Key Contributions and Results

1. Existence of Non-Residually Finite Irreducible Lattices (Theorem A)

The primary contribution is the construction of five finite triangle complexes $Y^2_1, Y^3_1, Y^3_2, Y^3_3, Y^3_4$ .

Their universal covers $X^q_i$ are exotic $\tilde{C}_2$ -buildings of thickness $q+1$ (thickness 3 for $q=2$ , thickness 4 for $q=3$ ).
Their fundamental groups $\Gamma^q_i = \pi_1(Y^q_i)$ are uniform lattices that are not residually finite.
Specifically, the finite residual $\check{\Gamma}^q_i = (\Gamma^q_i)^{(\infty)}$ $\overset{ˇ}{Γ}_{i}^{q} = (Γ_{i}^{q})^{(\infty)}$ has finite index in $\Gamma^q_i$ $Γ_{i}^{q}$ .
- For $q=2$ , $\Gamma^2_1$ is its own finite residual (i.e., it has no proper finite index subgroups).
- For $q=3$ , the indices $[\Gamma^3_i : \check{\Gamma}^3_i]$ are 4 for $i=1,2$ and 8 for $i=3,4$ .

2. Simple CAT(0) Groups with Property (T) (Corollary B)

By combining the main theorem with previous results (specifically [LW26]), the authors deduce that the finite residuals $\check{\Gamma}^q_i$ are simple groups.

These groups are CAT(0) groups (as they act properly and cocompactly on CAT(0) spaces).
They possess Kazhdan's Property (T). The authors verify Property (T) for $\bar{\Gamma}^2_1$ using Ozawa's method and semidefinite programming, establishing a Kazhdan constant of at least $0.4147$.
Crucially, these groups are not quasi-isometric to a proper direct product. This distinguishes them from the known non-residually finite lattices on products of trees, which are quasi-isometric to a product of trees.

3. Rigidity and Non-Isomorphism

The authors prove that the buildings $X^q_i$ are pairwise non-isomorphic and that the lattices $\Gamma^q_i$ are pairwise non-quasi-isometric.

This is achieved by reconstructing finite balls in the Cayley graphs and showing that the resulting quotients $\check{\Gamma}^q_i \backslash X^q_i$ are non-isomorphic.
The automorphism groups of these buildings are discrete, confirming their "exotic" nature.

4. Classification of $\tilde{A}_2$ -lattices (Theorem C and Theorem 6.1)

The paper provides a complete classification of type-preserving, vertex-regular lattices on $\tilde{A}_2$ -buildings of thickness 3.

There are exactly 13 such lattices (up to equivariance).
Arithmetic Examples: Six of these are arithmetic. Notably, one arithmetic example (associated with $\Gamma_6$ ) is defined over the field $K = \mathbb{Q}(\sqrt{-23})$ and is not commensurable to previously studied lattices. The authors provide an explicit embedding of this lattice into a general unitary group $GU(f)(\mathbb{Q}_2)$ .
Exotic Examples: The remaining seven lattices act on exotic buildings. The authors analyze their automorphism groups and quotients, identifying some as perfect groups with no small quotients.

Significance

The paper claims to provide the first known examples of non-residually finite lattices on irreducible buildings. This resolves a conjecture that such objects exist but had remained unverified.

The significance extends to several areas:

Group Theory: It produces the first known simple CAT(0) groups with Property (T) that are not quasi-isometric to a direct product. This challenges the intuition that Property (T) and simplicity in this context might be linked to reducible structures.
Geometric Rigidity: The results demonstrate that non-residually finite behavior is not exclusive to reducible buildings (products of trees) but occurs in irreducible settings, expanding the landscape of known lattice behaviors.
Classification: The exhaustive classification of $\tilde{A}_2$ -lattices of thickness 3 provides a concrete dataset for understanding the relationship between arithmetic and exotic lattices, revealing new arithmetic examples and a rich variety of exotic structures.

The authors emphasize that the construction relies on computer-assisted search and verification, particularly for establishing the non-residual finiteness (by checking that specific elements lie in the finite residual) and for computing automorphism groups of large balls. The code and data are made available to support these verifications.

Technical Summary: Non-residually finite C~2\tilde{C}_2C~2​-lattices