arxiv: 2604.22711 · v1 · submitted 2026-04-24 · 🧮 math.NT · math.GR· math.RT

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Asymptotic behaviour of analytic torsion and cohomological torsion for mathbb{Q}-rank 1 arithmetic groups

Tim Berland

Authors on Pith no claims yet

Pith reviewed 2026-05-08 09:56 UTC · model grok-4.3

classification 🧮 math.NT math.GRmath.RT

keywords analytic torsioncohomological torsioncongruence subgroupsarithmetic groupsQ-rank 1asymptoticsSL(2)SO(n,1)

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The pith

Refined asymptotics of analytic torsion extend to congruence subgroups of a large family of reductive groups.

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

The paper extends previous results on the asymptotic behavior of analytic torsion for congruence subgroups of SL(n) to a wider class of reductive groups. This extension is then used to derive new asymptotics and upper bounds for the growth of torsion in the cohomology of congruence subgroups of SL(2) over the ring of integers in a number field. The same approach yields results for congruence subgroups of SO(n,1) where n is odd. A sympathetic reader would care because these bounds help understand the size of torsion subgroups in the cohomology of arithmetic manifolds as the congruence level varies.

Core claim

We extend the refined asymptotics of analytic torsion associated to congruence subgroups of SL(n) in previous work, to congruence subgroups in a large family of reductive groups. This is applied to give new asymptotics and bounds on the growth of torsion in the cohomology of congruence subgroups of SL(2,O_F) for F a number field, and of congruence subgroups in SO(n,1) with n odd.

What carries the argument

The refined asymptotics of analytic torsion for congruence subgroups, extended to Q-rank 1 arithmetic groups via their reductive structure.

If this is right

Analytic torsion asymptotics hold for the extended family of reductive groups.
Asymptotics for the growth of cohomological torsion in congruence subgroups of SL(2, O_F).
Upper bounds on the torsion growth in those cohomologies as the level increases.
Similar asymptotics and bounds apply to congruence subgroups in SO(n,1) for odd n.

Where Pith is reading between the lines

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

The same extension technique could be tested on other low Q-rank groups not explicitly covered here.
The bounds may help estimate torsion contributions in regulators or class numbers for these arithmetic groups.
Numerical checks for small n or small number fields could confirm the rate of growth in practice.

Load-bearing premise

The same technical conditions on the groups and congruence subgroups that enabled the SL(n) case continue to be satisfied in the larger family.

What would settle it

A direct calculation of analytic torsion for a low-level congruence subgroup in SL(2, O_F) for a quadratic field F and comparison to the predicted asymptotic growth rate.

read the original abstract

We extend the refined asymptotics of analytic torsion associated to congruence subgroups of $\operatorname{SL}(n)$ in previous work, to congruence subgroups in a large family of reductive groups. This is applied to give new asymptotics and bounds on the growth of torsion in the cohomology of congruence subgroups of $\operatorname{SL}(2,\mathcal{O}_F)$ for $F$ a number field, and of congruence subgroups in $\operatorname{SO}(n,1)$ with $n$ odd.

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.

Desk Editor's Note private letter to a colleague

Berland extends the analytic torsion asymptotics from SL(n) to other Q-rank 1 reductive groups by verifying the hypotheses and then applies it to bound torsion growth in two families.

read the letter

This paper's main contribution is carrying over the refined asymptotics for analytic torsion of congruence subgroups from the SL(n) setting to a wider collection of reductive groups with Q-rank 1. It does so by confirming that these groups satisfy the required conditions on their symmetric spaces, congruence filtrations, and spectral properties from the earlier work. The author then uses this to obtain new asymptotics and bounds for the growth of torsion in the cohomology of congruence subgroups of SL(2, O_F) where F is a number field, and for SO(n,1) with n odd. The paper does a good job making the extension explicit and showing how the applications follow from the standard comparison between analytic and Reidemeister torsion. The bounds are quantitative, which adds value for anyone tracking these growth rates in open problems. The limitation is that this is primarily an extension and application rather than a new conceptual framework. The core technique is the same as before, so the advance is in the scope of groups covered. If the hypothesis checks are straightforward, the paper is reliable but not transformative. There are no obvious circularities or unverified steps based on the description. The work stays within the technical area of arithmetic groups and their invariants. This is aimed at researchers focused on analytic torsion, Reidemeister torsion, and the cohomology of arithmetic manifolds. A general number theorist or topologist might find it too specialized. I recommend sending it to peer review. It provides concrete new results that can be verified and built upon in the field.

Referee Report

0 major / 2 minor

Summary. The paper extends the refined asymptotics of analytic torsion for congruence subgroups of SL(n) from prior work to congruence subgroups of a larger family of reductive groups with Q-rank 1. It verifies that these groups (including SL(2, O_F) for number fields F and SO(n,1) with n odd) satisfy the necessary structural hypotheses on symmetric spaces, congruence filtrations, and spectral properties, then derives new asymptotics and bounds on the growth of torsion in the cohomology of these congruence subgroups via the standard comparison of analytic and Reidemeister torsion.

Significance. If the extension holds, the work meaningfully broadens the applicability of refined analytic torsion asymptotics to additional arithmetic groups of Q-rank 1, yielding concrete new bounds on cohomological torsion growth for SL(2) over number fields and for odd-dimensional hyperbolic arithmetic groups. This strengthens the link between analytic torsion and arithmetic cohomology in settings beyond the SL(n) case, with the hypothesis-verification approach providing a clear, falsifiable route for further extensions.

minor comments (2)

[§2.3] §2.3: the statement that the groups satisfy the spectral gap hypotheses of the prior SL(n) work would be clearer if it included a one-sentence summary of the key eigenvalue estimates used for each new family (SO(n,1) and SL(2,O_F)).
[Applications] The comparison between analytic and Reidemeister torsion in the applications section assumes the standard isomorphism without citing the precise reference for the Q-rank 1 case; adding the citation would improve traceability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were listed in the report, so we have no individual points requiring point-by-point rebuttal or manuscript changes at this stage. We have re-checked the manuscript for clarity and completeness and believe it is in final form, but we remain open to addressing any unlisted minor issues if the editor or referee provides further details.

Circularity Check

0 steps flagged

No significant circularity; extension via hypothesis verification

full rationale

The paper's central derivation extends refined analytic torsion asymptotics from the SL(n) case by explicitly confirming that the larger family of Q-rank 1 groups (including SL(2, O_F) and SO(n,1) with n odd) satisfy the structural hypotheses on symmetric spaces, congruence subgroup filtrations, and spectral properties required by the prior work. The applications to bounds on cohomological torsion growth then follow directly from the main theorem via the standard comparison between analytic and Reidemeister torsion. No load-bearing step reduces by construction to a fitted input, self-definition, or unverified self-citation chain; the verification supplies independent content, rendering the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review prevents enumeration of specific free parameters or invented entities; the work appears to rest on standard assumptions from the theory of reductive groups, analytic torsion, and cohomology of arithmetic groups.

pith-pipeline@v0.9.0 · 5375 in / 1228 out tokens · 29182 ms · 2026-05-08T09:56:03.899912+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 1 canonical work pages

[1]

Abert, N

[ABFG25] M. Abert, N. Bergeron, M. Frączyk, and D. Gaboriau,On homology torsion growth, J. Eur. Math. Soc.27(2025), no. 6, 2293–2357. [AGMY20] A. Ash, P. E. Gunnells, M McConnell, and D. Yasaki,On the growth of torsion in the cohomology of arithmetic groups, J. Inst. Math. Jussieu19(2020), no. 2, 537–569. [ARS12] P. Albin, F. Rochon, and D. Sher,Resolvent...

2025
[2]

Ash,Galois representations attached to modpcohomology ofGL(n,Z), Duke Math

[Ash92] A. Ash,Galois representations attached to modpcohomology ofGL(n,Z), Duke Math. J. 65(1992), no. 2, 235–255. [Ber25] T. Berland,Asymptotics of analytic torsion for congruence quotients ofSL(n,R)/SO(n), preprint, arXiv:2507.10339(2025). [BM83] D. Barbasch and H. Moscovici,L 2-index and the trace formula, J. Funct. An52(1983), no. 2, 151–201. [BV13] ...

work page arXiv 1992
[3]

Finis and E

[FL16] T. Finis and E. Lapid,On the continuity of the geometric side of the trace formula, Acta Math. Vietnam41(2016), no. 3, 425–455. [FL17] ,On the analytic properties of intertwining operators i: global normalizing factors, Bull. Iranian Math. Soc.43(2017), no. 4, 235–

2016
[4]

Math.230(2019), no

[FL19] ,On the analytic properties of intertwining operators ii: Local degree bounds and limit multiplicities, Israel J. Math.230(2019), no. 2, 771–

2019
[5]

Finis, E

[FLM15] T. Finis, E. Lapid, and W. Müller,Limit multiplicities for principal subgroups ofGL(n)and SL(n), J. Inst. Math. Jussieu14(2015), no. 3, 589–638. [Fra98] J. Franke,Harmonic analysis in weightedL 2-spaces, Ann. Sci. Ecole Norm. Sup.4e serie, t. 31(1998), 181–279. [Kac90] V. G. Kac,Infinite dimensional Lie algebras, 3 ed., Cambridge University Press,

2015
[6]

Lott,Heat kernels on covering spaces and topological invariants, J

[Lot92] J. Lott,Heat kernels on covering spaces and topological invariants, J. Differential Geom.7 (1992), 471–510. [Lüc16] W. Lück,Survey on approximatingL 2-invariants by their classical counterparts: Betti num- bers, torsion invariants and homological growth., EMSS3(2016), 269–344. [Mat15] J. Matz,Bounds for global coefficients in the fine geometric ex...

1992
[7]

Wang,Dimension of a minimal nilpotent orbit, Proc

[Wan99] W. Wang,Dimension of a minimal nilpotent orbit, Proc. Am. Math. Soc.127(1999), no. 3, 935–936. Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark Email address:twb@math.ku.dk

1999