arxiv: 2509.00857 · v3 · submitted 2025-08-31 · 🧮 math.GT

On Systole, Kissing Number and Volume of Arithmetic Manifolds

Plinio Guillel Pino Murillo This is my paper

Pith reviewed 2026-05-18 19:41 UTC · model grok-4.3

classification 🧮 math.GT

keywords arithmetic manifoldssystolekissing numberhyperbolic manifoldsarithmetic groupslocally symmetric spacesvolume

0 comments p. Extension

The pith

Arithmetic manifolds supply the constructions that answer how systole and kissing number grow with volume in hyperbolic manifolds.

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

The paper introduces arithmetic groups and the arithmetic manifolds they produce by using them as the source of answers to two geometric questions on the growth of systole and kissing number in hyperbolic manifolds. It works out the precise answers for these questions in dimension two. The exposition then surveys what is known in higher dimensions and for other locally symmetric spaces before listing open questions.

Core claim

Arithmetic manifolds, obtained as quotients of symmetric spaces by arithmetic subgroups, furnish the explicit examples that resolve the growth questions for systole and kissing number in hyperbolic manifolds, with full details given in dimension two and partial information supplied for higher dimensions and other symmetric spaces.

What carries the argument

Arithmetic manifold, a quotient space formed by an arithmetic subgroup acting on a symmetric space, which supplies the controlled geometric examples needed to track systole length and kissing number relative to volume.

If this is right

In dimension two the systole and kissing number of arithmetic hyperbolic surfaces are determined explicitly in terms of volume.
In higher dimensions arithmetic hyperbolic manifolds still give the best known growth controls, though the picture is incomplete.
The same arithmetic technique produces analogous bounds for systole and kissing number on other locally symmetric spaces.
The constructions link the geometric invariants directly to the volume of the fundamental domain.

Where Pith is reading between the lines

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

If arithmetic manifolds achieve the extremal growth rates, then number-theoretic properties of the defining groups would govern the geometry of all large-volume hyperbolic manifolds.
The open questions raised at the end suggest testing whether non-arithmetic examples can ever match or exceed the arithmetic growth rates.
The same arithmetic approach might be applied to control other invariants such as the length spectrum or the injectivity radius in the same manifolds.

Load-bearing premise

The geometric questions on systole and kissing number growth have been resolved primarily through arithmetic constructions.

What would settle it

Discovery of a non-arithmetic hyperbolic manifold whose systole grows faster than every arithmetic example of comparable volume would show that arithmetic constructions do not give the full picture.

read the original abstract

The purpose of this expository article is to give a down-to-hearth introduction to the notion of an arithmetic group and arithmetic manifold. To achieve this we have decided to bring two geometrical questions relating the growth of systole and kissing number in hyperbolic manifolds, as a motivational guide, whose answers so far are given by the use of arithmetic manifolds. As a consequence, we answer these questions in detail for dimension 2, we mention was is known for hyperbolic manifolds of higher dimension, and also for other locally symmetric spaces. We end the exposition with some open questions.

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

This is a clear expository piece that introduces arithmetic manifolds by walking through known answers to systole and kissing number questions, with solid detail in dimension 2 but no new theorems.

read the letter

The main takeaway is that this paper is an expository introduction to arithmetic groups and manifolds, framed around two standard questions in hyperbolic geometry: how systoles grow and what happens with kissing numbers. It gives detailed answers for dimension 2 and surveys the higher-dimensional and other locally symmetric cases without claiming original results. That framing works as a motivational hook and keeps the exposition grounded in concrete geometry rather than abstract definitions alone. The dimension-2 section stands out because it actually spells out the constructions and how they resolve the questions, which makes the material more usable for someone trying to see the link between arithmetic examples and geometric bounds. The survey parts are concise overviews of what is already in the literature, which is appropriate for this type of article. The open questions at the end are the usual ones and do not feel forced. One minor soft spot is that readers already comfortable with the arithmetic side or the geometric questions will mostly see a reorganization of known facts rather than fresh insight. The paper does not overstate its scope, so this is not a serious flaw, just the natural limit of an expository review. It is aimed at graduate students or researchers who want a motivated entry point into arithmetic manifolds from a hyperbolic geometry perspective. Someone preparing a reading group or looking for a clear summary of the current state would find it useful. The work shows clear thinking and honest engagement with the literature, so it deserves a serious referee even though it is not original research. I would recommend sending it to peer review for a venue that publishes expository articles.

Referee Report

0 major / 2 minor

Summary. The manuscript is an expository article that introduces the notions of arithmetic groups and arithmetic manifolds. It motivates the exposition by recalling two standard questions on the growth of systole and kissing number for hyperbolic manifolds, notes that known positive answers rely on arithmetic constructions, supplies detailed answers in dimension 2, surveys what is known in higher dimensions and for other locally symmetric spaces, and closes with open questions.

Significance. If the exposition accurately presents the established literature, the paper offers a readable entry point that connects two classical geometric problems to the arithmetic constructions that resolve them. The explicit focus on dimension 2 and the survey of higher-dimensional and non-hyperbolic cases could be useful for readers entering the area; the absence of new theorems is consistent with the stated expository purpose.

minor comments (2)

The abstract states that the paper 'answers these questions in detail for dimension 2'; a brief sentence in the introduction clarifying that this means a self-contained exposition of the known arithmetic constructions (rather than new proofs) would prevent any reader from misinterpreting the scope.
Section headings and the final open-questions paragraph would benefit from explicit cross-references to the specific theorems or constructions discussed earlier, to help readers locate the relevant arithmetic examples quickly.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and encouraging report. We are pleased that the expository purpose, the detailed treatment in dimension 2, and the survey of higher-dimensional cases were viewed as useful for readers entering the area.

Circularity Check

0 steps flagged

Expository survey with no load-bearing derivation reducing to self-inputs

full rationale

The paper is framed as an expository introduction to arithmetic groups and manifolds, using two standard geometric questions (systole and kissing-number growth) as motivation. It states that answers to these questions are already furnished by arithmetic constructions in the literature, then reviews those known results in detail for dimension 2 and surveys higher dimensions. No new theorems, predictions, or derivations are claimed that could reduce by construction to fitted parameters, self-citations, or ansatzes internal to the paper. All central claims rest on externally established facts rather than any self-referential chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

This expository article relies on standard definitions and results from hyperbolic geometry and arithmetic group theory without introducing new free parameters, ad hoc axioms, or invented entities.

axioms (1)

standard math Established properties of arithmetic groups and hyperbolic manifolds from prior literature
The exposition builds directly on classical results in the field.

pith-pipeline@v0.9.0 · 5613 in / 1107 out tokens · 40300 ms · 2026-05-18T19:41:13.450529+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

answers to questions relating the growth of systole and kissing number in hyperbolic manifolds are given by the use of arithmetic manifolds

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages · 1 internal anchor

[1]

Systoles of hyperbolic 4-manifolds

I. Agol, Systoles of hyperbolic 4-manifolds, arXiv preprint math/0612290 (2006)

work page internal anchor Pith review Pith/arXiv arXiv 2006
[2]

Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Annals of mathematics (1962), 485--535

A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Annals of mathematics (1962), 485--535

work page 1962
[3]

M. F. Bourque and B. Petri, Kissing numbers of closed hyperbolic manifolds, American Journal of Mathematics 144 (2022), no. 4, 1067--1085

work page 2022
[4]

Buser and P

P. Buser and P. Sarnak, On the period matrix of a Riemann surface of large genus (with an appendix by J. H. Conway and N. J. A. Sloane) , Inventiones Mathematicae 117 (1994), 27--56

work page 1994
[5]

M. V. Belolipetsky and S. A Thomson, Systoles of hyperbolic manifolds, Algebraic & Geometric Topology 11 (2011), no. 3, 1455--1469

work page 2011
[6]

3, 372--377

S Chowla, J Cowles, and M Cowles, On the number of conjugacy classes in SL (2, Z ) , Journal of Number Theory 12 (1980), no. 3, 372--377

work page 1980
[7]

D \'o ria and P

C. D \'o ria and P. G. P Murillo, Hyperbolic 3-manifolds with large kissing number, Proceedings of the American Mathematical Society 149 (2021), no. 11, 4595--4607

work page 2021
[8]

D \'o ria, C

E. D \'o ria, C. Freire and P.G.P Murillo, Hyperbolic manifolds with a large number of systoles, Transactions of the American Mathematical Society 377 (2024), no. 02, 1247--1271

work page 2024
[9]

Emery, V

I. Emery, V. Kim and P. G. P. Murillo, On the systole growth in congruence quaternionic hyperbolic manifolds, Bulletin of the London Mathematical Society 54 (2022), no. 4, 1265--1281

work page 2022
[10]

Fanoni and H

F. Fanoni and H. Parlier, Systoles and kissing numbers of finite area hyperbolic surfaces, Algebraic & Geometric Topology 15 (2016), no. 6, 3409--3433

work page 2016
[11]

and Dória C., Closed geodesics on semi-arithmetic riemann surfaces, Mathematical Research Letters 29 (2022), no

Cosac G. and Dória C., Closed geodesics on semi-arithmetic riemann surfaces, Mathematical Research Letters 29 (2022), no. 4, 41

work page 2022
[12]

Gendulphe, Systole et rayon interne des vari \'e t \'e s hyperboliques non compactes , Geometry & Topology 19 (2015), no

M. Gendulphe, Systole et rayon interne des vari \'e t \'e s hyperboliques non compactes , Geometry & Topology 19 (2015), no. 4, 2039--2080

work page 2015
[13]

Guth and A

L. Guth and A. Lubotzky, Quantum error, correcting codes and 4-dimensional arithmetic hyperbolic manifolds , Journal of Mathematical Physics 55 (2014), no. 8

work page 2014
[14]

Gromov, Systoles and intersystolic inequalities , Institut des Hautes \'E tudes Scientifiques

M. Gromov, Systoles and intersystolic inequalities , Institut des Hautes \'E tudes Scientifiques. S \'e minaires & Congr \'e s 1 (1996)

work page 1996
[15]

Katok, Fuchsian groups, University of Chicago press, 1992

S. Katok, Fuchsian groups, University of Chicago press, 1992

work page 1992
[16]

2, 349--357

I Kim, Systole on locally symmetric spaces, Bulletin of the London Mathematical Society 52 (2020), no. 2, 349--357

work page 2020
[17]

M. Katz, M. Schaps, and U. Vishne, Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups , Journal of Differential Geometry 76 (2007), 399--422

work page 2007
[18]

G. S. Lakeland and C. J. Leininger, Systoles and dehn surgery for hyperbolic 3-manifolds, Algebraic & Geometric Topology 14 (2014), no. 3, 1441--1460

work page 2014
[19]

Lapan, S

B. Lapan, S. Linowitz and J. S. Meyer, Systole inequalities up congruence towers for arithmetic locally symmetric spaces, Communications in Analysis and Geometry 31 (2024), no. 4, 847--878

work page 2024
[20]

D. W. Morris, Introduction to Arithmetic Groups , Deductive Press, 2015

work page 2015
[21]

P. G. P. Murillo, On growth of systole along congruence coverings of hilbert modular varieties, Algebraic & Geometric Topology 17 (2017), no. 5, 2753--2762

work page 2017
[22]

3, 1083--1102

, Systole of congruence coverings of arithmetic hyperbolic manifolds, Groups, Geometry, and Dynamics 13 (2019), no. 3, 1083--1102

work page 2019
[23]

Parlier, Kissing numbers for surfaces, Journal of Topology 6 (2013), no

H. Parlier, Kissing numbers for surfaces, Journal of Topology 6 (2013), no. 3, 777--791

work page 2013
[24]

P. S. Schaller, Arithmetic fuchsian groups and the number of systoles, Mathematische Zeitschrift 223 (1996), no. 1, 13--25

work page 1996
[25]

1, 191--198

, Compact riemann surfaces with many systoles, Duke Mathematical Journal 84 (1996), no. 1, 191--198

work page 1996
[26]

, Extremal riemann surfaces with a large number of systoles, Contemporary Mathematics 201 (1996), 9--20

work page 1996
[27]

W. P. Thurston, The geometry and topology of 3-manifolds, Lecture Notes from Princeton University, 1978-80

work page 1978