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arxiv: 2312.06507 · v3 · submitted 2023-12-11 · 🧮 math.NT · math.CO· math.GR

Ramanujan Bigraphs

Shai Evra , Brooke Feigon , Kathrin Maurischat , Ori Parzanchevski This is my paper

Pith reviewed 2026-05-24 05:27 UTC · model grok-4.3

classification 🧮 math.NT math.COmath.GR
keywords Ramanujan bigraphsCayley bigraphsRamanujan conjectureBruhat-Tits treesSarnak-Xue density hypothesisbiregular graphscutoff phenomenonPU_3
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The pith

Explicit constructions yield infinite families of (p^3+1, p+1)-regular Ramanujan Cayley bigraphs for infinitely many primes p.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends constructions of Ramanujan graphs to bigraphs by providing explicit arithmetic families of (p^3+1, p+1)-regular Cayley bigraphs for infinitely many p. These arise as quotients of Bruhat-Tits trees by congruence subgroups of lattices in PU_3 over the p-adics. The Ramanujan property ties directly to the Ramanujan conjecture on that group, which the authors identify holds in some explicit cases and fails in others. Non-Ramanujan examples are still shown to meet the Sarnak-Xue density hypothesis. Combinatorial characterizations of Ramanujan bigraphs and biexpanders are given, along with proofs of cutoff for non-backtracking walks.

Core claim

We give explicit constructions of infinite families of (p^3+1,p+1)-regular Ramanujan Cayley bigraphs, for infinitely many p. Both the LPS graphs and our ones are arithmetic, arising as quotients of Bruhat-Tits trees by congruence subgroups of arithmetic lattices in a p-adic group, PGL_2(Q_p) for LPS and PU_3(Q_p) for us. In both cases the Ramanujan property relates to the Ramanujan Conjecture (RC) on the respective groups. But while for PGL_2 the RC holds unconditionally, this is not so in the case of PU_3. We find explicit cases where the RC does and does not hold, and use this to construct arithmetic non-Ramanujan Cayley bigraphs as well, and prove that nevertheless they satisfy the Sarnak

What carries the argument

Cayley bigraphs defined as quotients of the Bruhat-Tits tree of PU_3(Q_p) by congruence subgroups, with the Ramanujan property for bigraphs given by a spectral bound on the adjacency operator.

If this is right

  • The bigraphs satisfy a pseudorandomness characterization of Ramanujan bigraphs.
  • They exhibit the cutoff phenomenon with bounded window for non-backtracking random walks, either from the Ramanujan property or the density hypothesis.
  • Golden gates for PU_3 are obtained from the constructions.
  • Optimal strong approximation holds for p-arithmetic subgroups of PSU_3.
  • First Betti numbers vanish for Picard modular surfaces arising in the setting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The biexpander notion introduced for bigraphs may extend mixing time results to other biregular settings beyond the Ramanujan case.
  • When the conjecture fails, the density hypothesis still supplies quantitative control useful for approximation algorithms or coding theory applications.
  • Similar quotient constructions on other p-adic groups with conditional conjectures could produce additional families of graphs with controlled expansion.

Load-bearing premise

The Ramanujan property of these bigraphs is controlled by whether the Ramanujan conjecture holds for the group PU_3 over the p-adics.

What would settle it

For a small prime p where the construction is explicit, compute the largest eigenvalue of the adjacency matrix of the resulting bigraph and check whether it exceeds the Ramanujan bound derived from the conjecture.

Figures

Figures reproduced from arXiv: 2312.06507 by Brooke Feigon, Kathrin Maurischat, Ori Parzanchevski, Shai Evra.

Figure 1.1
Figure 1.1. Figure 1.1: The non-backtracking spectrum of some of the (p 3+1, p+1)-regular bigraphs X p,q E . These are Cayley bigraphs of P SL3(q) for q ≡ 1 (3), and of P SU3(q) for q ≡ 2 (3). They are adj-Ramanujan, but not Ramanujan: the eigenvalues ±i √ K = ±ip3/2 are neither trivial nor in the spectrum of the covering tree Tp3+1,p+1. 5 5 3 2 1 1 2 3 40 20 20 40 6 4 2 2 4 6 100 50 50 100 10 5 5 10 [PITH_FULL_IMAGE:figures/f… view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: The non-backtracking spectrum of some Ramanujan bigraphs X p,q G . These are (p 3+1, p+1)-regular Cayley bigraphs of P SL3(q) for q ≡ 1 (4) and of P SU3(q) for q ≡ 3 (4). The graphs we study are the quotients of the tree by congruence subgroups of such lattices. In order for them to be Ramanujan, the infinite-dimensional automorphic representations associated with these subgroups should have tempered loc… view at source ↗
Figure 1.3
Figure 1.3. Figure 1.3: A schematic view of section dependency. While Section 8 makes use of all previous ones, it can also be read beforehand, relying on the previous results as black boxes. Acknowledgement. The authors wish to thank Cristina Ballantine who helped initiate this project and made fundamental contributions to it. The authors thank Sug Woo Shin for pointing out a strengthening of our results in Section 7. This res… view at source ↗
Figure 1.4
Figure 1.4. Figure 1.4: The automorphic Hamantash depicts the spectrum of the Hecke operator A1 on an Ae2-complex; it is composed of the endoscopic crust Ep (the red curve), and the Ramanujan filling, which is the spectrum of A1 on the building of P GL3 (Qp) (the blue shade). The dots show the nontrivial A1-spectrum of a few X p,q E Cayley complexes, with the underlying group Gq specified. Our next application (Section 8.4) is … view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: Left: Two Cayley bigraphs arising from involutions in G = Sym(3), as in Ex￾ample 2.5. In black and red is the (2, 2)-bigraph obtained from S = {(1 2),(2 3)}, with the left (resp. right) vertices drawn in red (resp. black); the dashed edges are those of the “original” Cayley graph Cay(G, S). Adding the blue edges and vertices gives the (3, 2)-bigraph obtained from S = {(1 2),(2 3),(1 3)}, and now the righ… view at source ↗
read the original abstract

In their seminal paper, Lubotzky, Phillips and Sarnak (LPS) defined the notion of regular Ramanujan graphs and gave an explicit construction of infinite families of $(p+1)$-regular Ramanujan Cayley graphs, for infinitely many primes $p$. In this paper we extend the work of LPS and its successors to bigraphs (biregular bipartite graphs), in several aspects: we investigate the combinatorial properties of various generalizations of the notion of Ramanujan graphs, define a notion of Cayley bigraphs, and give explicit constructions of infinite families of $(p^3+1,p+1)$-regular Ramanujan Cayley bigraphs, for infinitely many $p$. Both the LPS graphs and our ones are arithmetic, arising as quotients of Bruhat-Tits trees by congruence subgroups of arithmetic lattices in a $p$-adic group, $PGL_2(\mathbb{Q}_p)$ for LPS and $PU_3(\mathbb{Q}_p)$ for us. In both cases the Ramanujan property relates to the Ramanujan Conjecture (RC) on the respective groups. But while for $PGL_2$ the RC holds unconditionally, this is not so in the case of $PU_3$. We find explicit cases where the RC does and does not hold, and use this to construct arithmetic non-Ramanujan Cayley bigraphs as well, and prove that nevertheless they satisfy the Sarnak-Xue density hypothesis. On the combinatorial side, we present a pseudorandomness characterization of Ramanujan bigraphs, and a more general notion of biexpanders. We show that our Ramanujan bigraphs exhibit the cutoff phenomenon with bounded window size for non-backtracking random walks, either as a consequence of the Ramanujan property, or of the Sarnak-Xue density hypothesis. Finally, we present some other applications of our work: golden gates for $PU_3$, Ramanujan and non-Ramanujan complexes of type $\widetilde{A}_2$, optimal strong approximation for $p$-arithmetic subgroups of $PSU_3$ and vanishing of first Betti numbers of Picard modular surfaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper extends the LPS construction of Ramanujan graphs to the setting of bigraphs. It defines Cayley bigraphs and provides explicit arithmetic constructions, via quotients of the Bruhat-Tits tree of PU_3(Q_p) by congruence subgroups, of infinite families of (p^3+1, p+1)-biregular Ramanujan Cayley bigraphs for infinitely many primes p. It identifies explicit p where the Ramanujan conjecture holds or fails for the relevant spherical representations of PU_3, constructs corresponding non-Ramanujan examples that still satisfy the Sarnak-Xue density hypothesis, gives a pseudorandomness characterization of Ramanujan bigraphs, proves cutoff for non-backtracking walks, and lists applications including golden gates, A_2 complexes, strong approximation, and vanishing Betti numbers.

Significance. If the infinitude claim is established unconditionally, the work supplies the first explicit infinite families of Ramanujan bigraphs in this biregular setting and gives concrete arithmetic examples where the Ramanujan conjecture for PU_3 can be verified or falsified. The combinatorial characterizations, cutoff results under either the Ramanujan property or the density hypothesis, and the explicit non-Ramanujan yet density-satisfying examples are of independent interest for expander theory and random walks on groups.

major comments (2)
  1. [Abstract and §1] Abstract and §1: The central claim of 'explicit constructions of infinite families of (p^3+1,p+1)-regular Ramanujan Cayley bigraphs, for infinitely many p' is load-bearing. The Ramanujan property is equivalent to the Ramanujan conjecture holding for the relevant representations of PU_3(Q_p). The manuscript states that explicit cases exist where the conjecture holds and where it fails, but supplies no general argument (density, arithmetic progression, or unconditional proof) establishing infinitely many such good p. This leaves the infinitude assertion unsupported, unlike the LPS case for PGL_2 where the conjecture is known for every prime.
  2. [§4] §4 (construction of the Cayley bigraphs) and the statement of the main theorem: The explicit families are obtained only for those p where the Hecke eigenvalues lie inside the Ramanujan bound. Without either a proof that there are infinitely many such p or an explicit infinite list (rather than a finite verified list), the theorem as stated does not deliver the claimed infinite families of Ramanujan bigraphs.
minor comments (2)
  1. [§2] Notation for the biregular parameters (p^3+1, p+1) is introduced without an early clarifying sentence relating it to the local degree of the Bruhat-Tits tree of PU_3.
  2. [§3] The pseudorandomness characterization of Ramanujan bigraphs in §3 would benefit from an explicit comparison table with the classical LPS Ramanujan-graph characterization.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of significance, and constructive major comments. We agree that the infinitude claim requires clarification and will revise the manuscript accordingly. Below we respond point-by-point to the two major comments.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1: The central claim of 'explicit constructions of infinite families of (p^3+1,p+1)-regular Ramanujan Cayley bigraphs, for infinitely many p' is load-bearing. The Ramanujan property is equivalent to the Ramanujan conjecture holding for the relevant representations of PU_3(Q_p). The manuscript states that explicit cases exist where the conjecture holds and where it fails, but supplies no general argument (density, arithmetic progression, or unconditional proof) establishing infinitely many such good p. This leaves the infinitude assertion unsupported, unlike the LPS case for PGL_2 where the conjecture is known for every prime.

    Authors: We agree that, unlike the PGL_2 case, the Ramanujan conjecture for the relevant spherical representations of PU_3(Q_p) is not known unconditionally for infinitely many p, and the manuscript supplies no density or arithmetic-progression argument establishing this. The paper verifies the conjecture explicitly for several specific primes p (yielding Ramanujan bigraphs) and exhibits primes where it fails (yielding non-Ramanujan examples that still satisfy the Sarnak-Xue density hypothesis). For each fixed such p the congruence-subgroup quotients produce infinite families. The abstract and §1 do, however, overstate the claim by asserting constructions 'for infinitely many p' without sufficient qualification. We will revise the abstract, introduction, and related statements to clarify that the arithmetic constructions are available precisely for those p at which the relevant Ramanujan conjecture holds, that we provide explicit verified examples of such p, and that the infinitude of good p remains open (tied to the conjecture for PU_3). This removes the load-bearing unsupported assertion while preserving the explicit constructions and the contrast with LPS. revision: yes

  2. Referee: [§4] §4 (construction of the Cayley bigraphs) and the statement of the main theorem: The explicit families are obtained only for those p where the Hecke eigenvalues lie inside the Ramanujan bound. Without either a proof that there are infinitely many such p or an explicit infinite list (rather than a finite verified list), the theorem as stated does not deliver the claimed infinite families of Ramanujan bigraphs.

    Authors: We agree. The main theorem in §4 is formulated for primes p at which the Hecke eigenvalues satisfy the Ramanujan bound. The manuscript supplies a finite list of explicitly verified such p (together with counterexamples where the bound fails). No unconditional proof or infinite explicit list of good p is given. We will revise the statement of the theorem, the surrounding discussion in §4, and any cross-references to make this dependence explicit, to note that the construction yields infinite families for each verified good p, and to record that the infinitude question is open. The revised theorem will accurately reflect what is proved. revision: yes

Circularity Check

0 steps flagged

No circularity: constructions and RC analysis are independent of fitted inputs or self-referential definitions

full rationale

The paper constructs explicit arithmetic Cayley bigraphs as quotients of Bruhat-Tits trees by congruence subgroups of lattices in PU_3(Q_p), with the Ramanujan property defined via the external Ramanujan Conjecture on spherical representations of that group. Explicit cases where the conjecture holds or fails are identified separately, enabling both Ramanujan and non-Ramanujan examples plus the Sarnak-Xue hypothesis. This chain relies on standard arithmetic lattice theory and external RC status (known unconditionally only for PGL_2), not on redefining quantities in terms of themselves, renaming fitted parameters as predictions, or load-bearing self-citations whose content reduces to the present work. The infinitude claim for good p is presented as following from the explicit constructions where RC holds; no derivation step reduces by construction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on arithmetic properties of p-adic unitary groups and conditional results from the Ramanujan conjecture for specific cases.

axioms (2)
  • domain assumption Ramanujan Conjecture holds for specific congruence subgroups of PU_3(Q_p)
    Used to certify the Ramanujan property of the bigraphs in explicit cases.
  • domain assumption Sarnak-Xue density hypothesis applies to the non-Ramanujan cases constructed
    Invoked to prove properties even when RC fails.

pith-pipeline@v0.9.0 · 5947 in / 1485 out tokens · 38795 ms · 2026-05-24T05:27:35.915308+00:00 · methodology

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