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Edge zeta function and closed cycles in the standard non-uniform complex from $\operatorname{PGL}_3$

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Pith's one-line read This paper proves an exact rational formula for the edge zeta function of the standard non-uniform quotient complex of the PGL(3) Bruhat–Tits building by the lattice PGL(3,F_q[t]), and derives from it a closed weighted count of closed…

desk verdict A genuinely new computation of an edge zeta function for a non-compact PGL3 quotient, with a repairable gap in the determinant-limit step. read the letter →

arxiv 2411.15489 v1 pith:CD3XUMBX submitted 2024-11-23 math.GR math.COmath.DSmath.GTmath.NTmath.RT

Soonki Hong , Sanghoon Kwon This is my paper · ORCID
classification math.GRmath.COmath.DSmath.GTmath.NTmath.RT MSC 37E1520E4222E5020G25
keywords edgezetafunctionweightedcomplexBruhat-TitsbuildingPGL(3)fieldclosedgeodesiccyclesIharatruncation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper defines an edge zeta function for weighted complexes—infinite quotients where each directed edge carries a weight counting how many geodesic continuations it has—and proves an exact rational formula for the edge zeta function of the standard non-uniform quotient $\Gamma\setminus\mathcal{B}$, where $\Gamma=\operatorname{PGL}(3,\mathbb{F}_q[t])$ and $\mathcal{B}$ is the Bruhat–Tits building of $\operatorname{PGL}(3,\mathbb{F}_q(\!(t^{-1})\!))$. The zeta function converges for $\operatorname{Re}(s)>2$ and equals a rational function in $q^{-s}$ with four factors. From that formula the paper extracts a closed count of weighted closed geodesic cycles: $N_m=3q^{6r}-6q^{4r}+3q^{3r}$ when $m=3r$, and $N_m=0$ otherwise. The result matters because it carries the Ihara–Bass determinant method from finite graphs and infinite trees into a genuinely higher-rank, non-compact building quotient.

What carries the argument

The engine of the proof is the operator $T_k$ on the vector space formally spanned by the oriented type-$k$ edges of $\Gamma\setminus\mathcal{B}$, with matrix entries $w(e,e')$ counting the number of lifts of $e'$ that continue a lift of $e$ along a geodesic without forming a chamber. A trace computation identifies $Z_{\Gamma,k}(u)^{-1}$ with $\det(I-u^kT_k)$. The determinant is evaluated by truncating the complex to the subcomplex $X_k$ of vertices with $n\le k$, applying block Gaussian elimination to the finite matrix $I-uT_{1,k}$, and taking the limit in a specific order; the recurrence for the Schur-complement blocks $B_{k,\ell}(u)$ produces $(1-q^3u^3)(1-q^6u^3)/(1-q^4u^3)^2$ for type 1 and the same expression in $u^2$ for type 2.

What would settle it

For $q=2$, enumerate all type-1 admissible cycles of length 3 in the quotient using the weight diagram of Section 3 and compute $\sum_c w(c)\ell(c_0)$; the formula predicts 120, so any other value would disprove the rational zeta formula. The same check can be done by computing $\operatorname{Tr}(T_1^3)$ directly from the operator.

Watch

Extended reading notes

Core claim

The central claim is that for $\Gamma=\operatorname{PGL}(3,\mathbb{F}_q[t])$, the edge zeta function is given by $$Z_\Gamma($q^{{-s}}$)=\frac{(1-$q^{{4-3s}}$)^2(1-$q^{{4-6s}}$)^2}{(1-$q^{{3-3s}}$)(1-$q^{{6-3s}}$)(1-$q^{{3-6s}}$)(1-$q^{{6-6s}}$)}$$ and converges for $\operatorname{Re}(s)>2$. The proof goes through the type-$k$ edge zeta functions $Z_{\Gamma,k}(q^{-s})=1/\det(I-q^{-sk}T_k)$, where $T_k$ is the weighted adjacency operator on oriented type-$k$ edges of the quotient. The determinant is evaluated by truncating the infinite complex in a horizontal direction, computing the finite determinants with a block-matrix recurrence, and taking the limit; type 2 is handled by the same computation with $u$ replaced by $u^2$. The same calculation yields the weighted closed-cycle count $N_m=3q^{6r}-6q^{4r}+3q^{3r}$ for $m=3r$ and $N_m=0$ otherwise.

Load-bearing premise

The load-bearing premise is that the determinant of the infinite operator $I-uT_1$ is the limit of the determinants of the finite truncated operators $I-uT_{1,k}$; the paper states this limit in Section 5.2 without proving the interchange, and the convergence lemma it proves covers only traces of powers of $T_k$, not determinants.

Editorial extensions

If this is right

  • The exact closed-cycle count follows directly: a geodesic cycle of type 1 contributes only when its geometric length is a multiple of 3, with $N_{3r}=3q^{6r}-6q^{4r}+3q^{3r}$.
  • The Euler product over primitive geodesic cycles converges in the half-plane $\operatorname{Re}(s)>2$, giving a well-defined zeta function for this non-compact quotient.
  • The full edge zeta function is the product of two identical rational factors, one for each edge type, confirming the type-1/type-2 symmetry of the building.
  • The determinant formula extends the Bass–Ihara relation to an infinite weighted complex in higher rank, where the operator $T_k$ is traceable although the complex is non-compact.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the truncation-limit computation is as robust as it appears, the same 'truncate in one direction, take Schur complements, then let the truncation grow' scheme should yield rational edge zeta functions for other non-cocompact lattices in $\operatorname{PGL}_n$ that admit a finitely described fundamental domain.
  • The count $N_m=0$ for $m$ not divisible by 3 is strong enough to test by direct enumeration of admissible cycles in the finite truncated graphs for small $q$; a single mismatch would localize exactly where the infinite-determinant limit fails.
  • A natural next step, already anticipated in the paper's closing remarks, is to define a chamber zeta function for this infinite quotient and test whether the edge-chamber zeta identity proved for finite complexes survives in the infinite setting.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. This paper defines a geometric edge zeta function for the non-uniform quotient X = PGL(3,F_q[t])\PGL(3,F_q((t^{-1})))/PGL(3,F_q[[t^{-1}]]) and computes it explicitly. The main theorem (Theorem 1.1) states that for Γ = PGL(3,F_q[t]), the zeta function Z_Γ(q^{-s}) converges for Re s > 2 and equals the rational function (1 - q^{4-3s})^2(1 - q^{4-6s})^2 / ((1 - q^{3-3s})(1 - q^{6-3s})(1 - q^{3-6s})(1 - q^{6-6s})). The proof proceeds by introducing truncated subcomplexes X_k, computing det(I - uT_{1,k}) by finite block-matrix manipulations, passing to k → ∞, and then using a symmetry to handle the type-2 operator. A corollary gives the weighted count of closed type-1 cycles: N_m = 3q^{6r} - 6q^{4r} + 3q^{3r} if m = 3r and 0 otherwise.

Significance. If the main formula is correct, this is a significant new result: it provides the first explicit rational zeta function for a non-compact higher-rank building quotient and a concrete cycle-counting statement, extending the finite-complex results of Kang–Li and the tree-lattice zeta functions of Deitmar–Kang. The computation is self-contained in the sense that the determinant formula is derived rather than assumed, and no parameters are fitted to the answer. The final expression is internally consistent with the log-derivative computation in the proof of Corollary 5.2. The principal weaknesses are the unproved interchange of truncation limits and the unproved type-2 symmetry; both are local and appear repairable within the scope of the paper.

major comments (3)
  1. [Section 5.2, Eq. (5.3)] The equality det(I − uT_1) = lim_{k→∞} det A'_k(u), where det A'_k(u) = lim_{N→∞} det M_{k,N}(u), is asserted without a theorem justifying either the N-limit or the k-limit for determinants of infinite operators. Lemma 4.4 establishes only that the series ∑ u^n T^n/n converges weakly and that T^n is traceable; it does not show that the truncated operators T_{1,k} converge to T_1 in a topology that preserves Fredholm determinants, nor that the two limits commute. Since formula (5.3), Theorem 5.1, and Corollary 5.2 are exact consequences of this limit interchange, the gap is load-bearing. A natural repair is to prove that every type-1 closed admissible cycle of length m is contained in X_k for all k ≥ m, so that Tr(T_{1,k}^m) = Tr(T_1^m) for k ≥ m, and then to pass coefficient-wise through det(I − uT) = exp(−∑_m Tr(T^m)u^m/m).
  2. [Section 5.3] The sentence 'Under the definition of T2,k on Yk, the matrix representation of I − uT2,k coincides with I − uT1,k' is stated without proof. The type-2 factor of the zeta function, and therefore the full product formula in Theorem 1.1, depend on this identification. The paper should provide an explicit bijection between the edge sets E_1(X_k) and E_2(Y_k) (or a direct comparison of the matrices) to justify this claim.
  3. [Section 4.3, Lemma 4.4] The proof of Lemma 4.4 asserts that there is α > 0 such that the series −∑_{n≥1} q^{-sn} T_k^n/n converges weakly and is traceable for |q^{-s}| < α, but no estimate on the growth of Tr(T_k^n) is given. Lemma 3.2 shows finiteness for each fixed n, not an exponential bound, and the positivity argument permits reordering sums but does not by itself imply convergence of the series. Since Proposition 4.5 and the subsequent determinant computations rely on this convergence, the lemma needs a proof or a reference for the required bound.
minor comments (6)
  1. [Abstract and Introduction] There are several typos that should be corrected: 'trunction', 'on e of', 'disceterete', 'weigthed', and 'postive'.
  2. [Corollary 5.2] The displayed formula reads N_m = 3q^{2r} − 6q^{4/3 r} + 3q^r if m = 3r, which is inconsistent with the proof just below it and with the abstract, both of which give N_m = 3q^{6r} − 6q^{4r} + 3q^{3r}.
  3. [Section 5.3] The phrase 'the operator T2,k on the set of oriented edges of color 1 in Yk' should presumably read 'type 2' rather than 'color 1'.
  4. [References] Reference [CM94] contains the garbled string 'W. M/suppress Lotkowski'; the author name should be corrected to W. Mlotkowski.
  5. [Figure 10] The label 'en,m−n,3n' appears to be a typo for 'en,m−n,3'.
  6. [Section 4.2] The definition of T2 is written as 'T2e = ∑_{e2→e1} w(e,e1)e1', which is confusing; the summation should be over same-type edges e' with s(e') = t(e), as in the definition of T1.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the zeta formula is derived from a self-contained trace/determinant computation, with only an ancillary self-citation that is not load-bearing.

full rationale

The paper's central result is not circular. The zeta function is defined as an Euler product over prime cycles, and the determinant identity Z_{Γ,k}(u)^{-1}=exp(Tr(log(1-uT_k))) is derived, not assumed, from the trace formula Tr(T_k^n)=Σ_{ℓ(c)=n} w(c)ℓ(c_0) in Lemma 4.2 and Proposition 4.3. The final rational expression is obtained by an explicit truncated-matrix computation in Sections 5.1 and 5.2, with no fitted parameters and no input data. The one self-citation, Lemma 3.1 citing [HK24], is not load-bearing: the lemma is established by reference to the external reduction theory of Prasad [Pr03], and the prior work by the present authors is used only as a 'see also' for the fundamental domain, not as the source of the zeta formula. The truncation-limit interchange det(I-uT_1)=lim_k det A'_k(u) is asserted without a full proof of convergence of Fredholm determinants; this is a genuine analysis gap that could affect the validity of Theorem 5.1, but it is a correctness or convergence issue, not a circularity: the desired formula is not built into the definition of the truncated determinants or into any fitted parameter. The type-2 determinant is reduced to type 1 by an asserted symmetry, which is a mathematical claim requiring verification rather than a circular definition. Therefore the appropriate circularity score is minimal.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper has no fitted free parameters. It relies on standard building theory, a cited reduction-theory lemma from the authors' earlier work, and the Cartwright-Lotkowski counting lemma. No new entities (particles, forces, dimensions) are introduced.

assumptions (4)
  • domain assumption Standard structure theory of Bruhat-Tits buildings for PGL3 over a local field, including the classification of vertices, edges, and apartments (Section 2).
    The paper uses standard facts about the building, such as the number of neighbors of a vertex (Lemma 2.1), without proof.
  • domain assumption Lemma 2.2, cited from Cartwright-Lotkowski [CM94], that there are q^2 geodesic extensions of a finite type-1 geodesic segment.
    This fact underlies the count of out-neighbors and the row sums of the edge operator.
  • domain assumption Lemma 3.1 on the reduction theory / fundamental domain for PGL(3,F_q[t]), cited from Prasad [Pr03] and the authors' previous paper [HK24].
    The fundamental domain of the quotient and the classification of vertices vm,n depend on this reduction theory.
  • domain assumption The trace-class convergence framework for infinite matrices (Section 4.3), i.e., that the operators T_k are traceable and the determinant can be expressed via the trace of log.
    The convergence of the Euler product to a determinant relies on the traceability argument, which is proved in the paper but uses positivity and boundedness assumptions; we list it as an assumption because the infinite determinant existence is not fully formalized.

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Pith. "Pith review of Edge zeta function and closed cycles in the standard non-uniform complex from $\operatorname{PGL}_3$." pith.science (2026). https://pith.science/paper/CD3XUMBX

@misc{pith2026241115489,
  author       = {Pith},
  title        = {Pith review of: Edge zeta function and closed cycles in the standard non-uniform complex from $\operatornamePGL_3$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CD3XUMBX}},
  note         = {Machine review of arXiv:2411.15489}
}
abstract

In this paper, we define the edge zeta function of weighted complex. We also present the formula for the edge zeta function of the standard non-uniform complex $\operatorname{PGL}(3,\mathbb{F}_q[t])\backslash\operatorname{PGL}(3,\mathbb{F}_q(\!(t^{-1})\!))/\operatorname{PGL}(3,\mathbb{F}_q[\![t^{-1}]\!])$, arising from the group $\operatorname{PGL}_3$, as a rational function. Applying trunction in a specific direction is one of the main ingredient. As a result, we obtain the exact formula for the number of closed cycles coming from geodesics in the building.

Figures

Figures reproduced from arXiv: 2411.15489 by the authors.

Figure 1
Figure 1. Local neighborhood of a vertex in B, q “ 2 2.2. Type of geodesics. Our goal is to define the geometric zeta function ZΓpq ´s q of a discrete subgroup Γ of PGLp3, Fq. For this, we need to decide what kind of closed geodesics to count. The 2-dimensional building B is obtained by gluing together its apartments along chambers. Each apartment is a Euclidean plane tiled by equilateral triangles. The Euclidean metric on ea… view at source ↗
Figure 2
Figure 2. Type of directed edges A path er1, er2, . . . , ern in B is called a geodesic path if it is a part of straight line in an apartment in B. Equivalently, it is a path with the condition that sperkq, speĄk`1q “ tperkq, tpeĄk`1q do not form a chamber in B for all 1 ď k ď n ´ 1. See [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Type 1 geodesic paths, allowed one and what is not Let a “ diagpt, 1, 1q P PGLp3, Fqppt ´1 qqq. Then, the group H given by H “ K X aKa´1 “ $ ’& ’% ¨ ˚˝ k11 k12 k13 k21 k22 k23 k31 k32 k33 ˛ ‹‚ P K : k21, k31 P t ´1Fqrrt ´1 ss , /. /- , called the parahoric subgroup of PGLp3, Fqppt ´1 qqq, is the stabilizer of the type 1 directed edge K Ñ aK in B. Thus, we may identify the set of type 1 directed edges with G{H. We ob… view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: The fundamental domain for ΓzB 3.2. Paths in the quotient that lift to geodesics. Our goal is to count the number of cycles that are obtained as quotients of geodesics in the universal cov￾ering. Let π : B Ñ PGLp3, FqrtsqzB be the natural projection map. A sequence of …
Figure 5
Figure 5. Figure 5: Local neighborhood of a vertex in B 1 q ´ 1 q 2 ´ q q 2 ´ q q 0 0 0 q 2 in Type 1 Type 2 out 1 q ´ 1 q 2 ´ q 0 q 2 ´ q q 0 0 q 2 [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: Local pattern of the weights wpe, e1 q for each type Now, by treating each edge as an alphabet and drawing an arrow from e to e 1 when wpe, e1 q is not zero, we create a large directed graph. This process is similar to representing a shift space as a directed graph [P…
Figure 7
Figure 7. Figure 7: Dots denote the edges of ΓzB q 2 q 2 ´ 1 q 2 ´ q q q ´ 1 1 [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: Global description of the weights w of out-neighbors We will conclude this section by proving the following lemma, which will be used in demonstrating the convergence of the zeta function. Lemma 3.2. For any positive integer n, there are finitely many edges e in X that…
Figure 9
Figure 9. Figure 9: (Geometric) length and algebraic length Define the out-neighbors of a type 1 edge pg1O3 , g2O3 q to be the type 1 edges pg2O3 , g2O3 q such that pg1O3 , g2O3 , g3O3 q is not a pointed chamber. Each type 1 edge has q 2 out-neighbors. Let E be the standard parahoric subg…
Figure 10
Figure 10. Figure 10: Labelling of type 1 the oriented edges [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: E1pX3q The operator T1,k on the set of oriented edges of color 1 in Xk is defined by T1,kem,n,1 :“ $ ’’& ’’% pq 2 ´ 1qe0,n,2 ` e0,n`1,1 if m “ 0 pq ´ 1qem,n,2 `pq 2 ´ qqem´1,n`1,3 ` em,n`1,1 if n ‰ 0 T1,kem,n,2 :“ $ ’’& ’’% pq 2 ´ qqem,0,3 ` qem`1,0,1 if n “ 0 pq 2 ´ …
Figure 12
Figure 12. Figure 12: and 13) fn,m´n,1 :“ vm,n Ñ vm`1,n`1 fn,m´n,2 :“ vm`1,n`1 Ñ vm`1,n fn,m´n,3 :“ vm`1,n Ñ vm,n. vm,n vm`1,n vm`1,n`1 fn,m´n,3 fn,m´n,1 fn,m´n,2 [PITH_FULL_IMAGE:figures/full_fig_p025_12.png]
Figure 13
Figure 13. Figure 13: E2pY3q [PITH_FULL_IMAGE:figures/full_fig_p025_13.png]

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    math.NT 2026-07 accept novelty 6.0 of 10

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