Actions of lattices in $S$-arithmetic groups on manifolds

Segev Gonen Cohen

arxiv: 2607.00697 · v1 · pith:W2BFP3J4new · submitted 2026-07-01 · 🧮 math.DS · math.GR· math.GT

Actions of lattices in S-arithmetic groups on manifolds

Segev Gonen Cohen This is my paper

Pith reviewed 2026-07-02 05:24 UTC · model grok-4.3

classification 🧮 math.DS math.GRmath.GT

keywords lattice actionsp-adic groupsS-arithmetic groupsC1 diffeomorphismscompact manifoldsfinitenessgroup actions

0 comments

The pith

Lattices in simple p-adic groups have only finite C1 actions on compact manifolds of dimension less than the rank.

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

The paper shows that actions by C1 diffeomorphisms of a lattice in a simple p-adic group on a compact manifold are finite whenever the manifold dimension is smaller than the rank of the group. It generalizes this to lattices in totally disconnected S-arithmetic groups, taking the critical dimension to be the largest rank among the simple factors involved. Sympathetic readers would be interested because the result limits the ways these groups can act smoothly and continuously on manifolds, forcing any such action to have finite image in the diffeomorphism group. The argument adapts techniques previously used for other groups to these p-adic and S-arithmetic settings.

Core claim

We prove that an action by C^1 diffeomorphisms of a lattice in a simple p-adic group on a compact manifold is finite, provided the dimension is less than the rank. We extend this statement to lattices in totally disconnected S-arithmetic groups, where the critical dimension is the maximal rank of the simple factors. This uses the machinery developed by Brown, Fisher, and Hurtado.

What carries the argument

The Brown-Fisher-Hurtado machinery for proving finiteness of group actions on manifolds, applied here to lattices in p-adic and S-arithmetic groups.

If this is right

Any such action on a manifold of dimension below the rank must have finite image.
The same finiteness conclusion holds for lattices in totally disconnected S-arithmetic groups when dimension is below the maximal rank of the simple factors.
The result gives no information on actions whose dimension meets or exceeds the critical rank value.
Finiteness follows directly once the dimension hypothesis and the applicability of the cited machinery are granted.

Where Pith is reading between the lines

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

The same approach might yield finiteness statements for lattices in other totally disconnected groups if the machinery can be verified for them.
It remains open whether non-finite actions appear precisely when dimension equals the critical rank.
The theorem constrains possible smooth realizations of these groups as subgroups of diffeomorphism groups in low dimensions.

Load-bearing premise

The Brown-Fisher-Hurtado machinery applies to lattices in simple p-adic groups and totally disconnected S-arithmetic groups to establish finiteness under the dimension condition.

What would settle it

A single non-finite C1 action of a lattice in a simple p-adic group on a compact manifold whose dimension is strictly less than the rank would disprove the claim.

read the original abstract

We prove that an action by $C^1$ diffeomorphisms of a lattice in a simple $p$-adic group on a compact manifold is finite, provided the dimension is less than the rank. We extend this statement to lattices in totally disconnected $S$-arithmetic groups, where the critical dimension is the maximal rank of the simple factors. This uses the machinery developed by Brown, Fisher, and Hurtado.

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

Extends BFH finiteness to p-adic and S-arithmetic lattices under the same dimension bound, but the adaptation of connected-group tools needs explicit verification.

read the letter

The paper proves that C^1 actions of lattices in simple p-adic groups on compact manifolds are finite when the manifold dimension is less than the rank, and extends the statement to lattices in totally disconnected S-arithmetic groups with the critical dimension being the max rank of the factors. This is the actual new content.

It takes the Brown-Fisher-Hurtado finiteness theorem and carries the conclusion over to these new classes of groups. The clean statement that keeps the same dimension threshold is useful for anyone tracking rigidity results in this area.

The soft spot is the reliance on BFH machinery. That work was built for connected real semisimple Lie groups and uses features like Cartan subgroups and connected-component dynamics on the derivative cocycle. P-adic groups are totally disconnected, so the paper has to show either that the original arguments still apply or that suitable replacements exist. The abstract says it uses the machinery, but without seeing the details of how the adaptation is handled, it is hard to judge whether the proof goes through.

This is for specialists working on group actions on manifolds and Zimmer-type rigidity. A reader already following BFH results will get a clear new statement to consider.

I would send it to peer review. Referees who know the original BFH paper can check whether the extension is properly justified or whether extra arguments are required for the p-adic case.

Referee Report

1 major / 0 minor

Summary. The manuscript proves that any C^1 action by diffeomorphisms of a lattice in a simple p-adic group on a compact manifold is finite provided the manifold dimension is strictly less than the rank; it extends the result to lattices in totally disconnected S-arithmetic groups, replacing the rank threshold by the maximal rank among the simple factors. The proof is obtained by invoking the Brown-Fisher-Hurtado finiteness theorem.

Significance. If the central claim holds, the result would constitute a genuine extension of smooth rigidity phenomena from connected real semisimple Lie groups to p-adic and totally disconnected settings, thereby enlarging the class of groups for which low-dimensional actions are forced to be finite.

major comments (1)

[Abstract / main argument] The manuscript invokes the Brown-Fisher-Hurtado theorem without an explicit check that its hypotheses are satisfied for lattices in simple p-adic groups. In particular, the original BFH argument relies on the existence of Cartan subgroups, real-rank dynamics on the derivative cocycle, and connectedness properties that are absent or require replacement in the totally disconnected topology of groups such as SL(n, Q_p). Because the entire finiteness statement rests on this applicability, the omission is load-bearing.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for pointing out the need for an explicit verification of the Brown-Fisher-Hurtado theorem's hypotheses in the p-adic setting. We will revise the paper to include this check.

read point-by-point responses

Referee: [Abstract / main argument] The manuscript invokes the Brown-Fisher-Hurtado theorem without an explicit check that its hypotheses are satisfied for lattices in simple p-adic groups. In particular, the original BFH argument relies on the existence of Cartan subgroups, real-rank dynamics on the derivative cocycle, and connectedness properties that are absent or require replacement in the totally disconnected topology of groups such as SL(n, Q_p). Because the entire finiteness statement rests on this applicability, the omission is load-bearing.

Authors: We agree with the referee that an explicit check is required and that the original BFH proof uses features specific to the real Lie group setting. In the revised version, we will insert a dedicated paragraph or subsection immediately following the statement of the main theorem, where we verify that the relevant hypotheses hold or have suitable analogues for lattices in simple p-adic groups. Key points include: (1) the role of Cartan subgroups is played by maximal split tori, which exist and act with the necessary hyperbolicity properties in the p-adic case; (2) the derivative cocycle dynamics can be analyzed using the p-adic valuation and the action on the projective space over the p-adic field; (3) while connectedness is absent, the proof relies on the existence of elements with contracting/expanding behavior on the tangent bundle, which is guaranteed by the rank assumption and the structure of S-arithmetic lattices. We will also note that the extension to totally disconnected S-arithmetic groups follows by reducing to the simple factors via the maximal rank condition. This revision will make the applicability transparent. revision: yes

Circularity Check

0 steps flagged

No circularity; result applies external BFH theorem to new groups

full rationale

The derivation consists of verifying that the Brown-Fisher-Hurtado finiteness theorem (cited as external machinery) applies to lattices in simple p-adic and totally disconnected S-arithmetic groups under the stated dimension condition. No self-citations, no fitted parameters renamed as predictions, no self-definitional steps, and no uniqueness theorems imported from the author's prior work. The central claim is an application of an independent result rather than a reduction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard facts about lattices, p-adic groups, manifold actions, and the applicability of the Brown-Fisher-Hurtado framework; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Lattices in simple p-adic groups admit actions by C1 diffeomorphisms on compact manifolds whose finiteness is governed by rank and dimension.
Invoked implicitly as the setting for the theorem in the abstract.
domain assumption The Brown-Fisher-Hurtado machinery extends without modification to the p-adic and S-arithmetic cases under the dimension bound.
Stated as the method used in the abstract.

pith-pipeline@v0.9.1-grok · 5585 in / 1336 out tokens · 36258 ms · 2026-07-02T05:24:05.421063+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 6 canonical work pages · 3 internal anchors

[1]

Extremal Lyapunov exponents: an invariance principle and applications

[AV10] A. Avila and M. Viana. “Extremal Lyapunov exponents: an invariance principle and applications”. en. In:Inventiones mathematicae181.1 (July 2010), pages 115–178. issn: 1432-1297. [BDZ22] A. Brown, D. Damjanovi´ c, and Z. Zhang. “C1 actions on manifolds by lattices in Lie groups”. en. In:Compositio Mathematica158.3 (Mar. 2022), pages 529–549.issn: 00...

work page arXiv 2010
[2]

Zimmer’s conjecture: Subexponential growth, measure rigidity, and strong property (T)

[BFH22] A. Brown, D. Fisher, and S. Hurtado. “Zimmer’s conjecture: Subexponential growth, measure rigidity, and strong property (T)”. In:Annals of Mathematics 196.3 (2022), pages 891–940. [BHV08] B. Bekka, P. de la Harpe, and A. Valette.Kazhdan’s Property (T). Volume

2022
[3]

Smooth ergodic theory ofZ d-actions

[BHW23] A. Brown, F. R. Hertz, and Z. Wang. “Smooth ergodic theory ofZ d-actions”. en. In:Journal of Modern Dynamics19.0 (May 2023). Publisher: Journal of Modern Dynamics, pages 455–540.issn: 1930-5311. [BM02] M. Burger and N. Monod. “Continuous bounded cohomology and applications to rigidity theory”. In:Geom. Funct. Anal.12.2 (2002), pages 219–280. [BM46...

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New York: Springer-Verlag, 1991.isbn: 978-0-387-97370-8

Graduate Texts in Mathematics. New York: Springer-Verlag, 1991.isbn: 978-0-387-97370-8. 26 [BP07] L. Barreira and Y. Pesin.Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents. Volume

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Entropy, Lyapunov exponents, and rigidity of group actions

[Br19] A. Brown. “Entropy, Lyapunov exponents, and rigidity of group actions”. en. In:Ensaios Matem´ aticos33.2019 (2019).issn: 21750432. [Br72] G. E. Bredon.Introduction to Compact Transformation Groups. Volume

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Progr` es r´ ecents concernant le programme de Zimmer (d’apr` es A. Brown, D. Fisher & S. Hurtado)

Pure and Applied Mathematics. New York and London: Academic Press, 1972.isbn: 978-0-12-128850-1. [Ca17] S. Cantat. “Progr` es r´ ecents concernant le programme de Zimmer (d’apr` es A. Brown, D. Fisher & S. Hurtado)”. In:S´ eminaire Bourbaki. 70 e ann´ ee, 2017–2018, no. 1136(2017), page

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[7]

Algebraic actions of discrete groups: thep-adic method

[CX18] S. Cantat and J. Xie. “Algebraic actions of discrete groups: thep-adic method”. In:Acta Mathematica220.2 (2018), pages 239–295. [DH20] B. Deroin and S. Hurtado.Non left-orderability of lattices in higher rank semi-simple Lie groups. arXiv:2008.10687 [math]. Aug

work page arXiv 2018
[8]

Diagonal Actions on Locally Homogeneous Spaces

[EL10] M. Einsiedler and E. Lindenstrauss. “Diagonal Actions on Locally Homogeneous Spaces”. In:Homogeneous flows, moduli spaces, and arithmetic. proceedings of the Clay Mathematics Institute Summer School. American Mathematical Society, 2010, pages 155–242. [EW11] M. Einsiedler and T. Ward.Ergodic Theory with a View Towards Number Theory. Volume

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Springer, 2011.isbn: 978 0 85729 020

Graduate Texts in Mathematics. Springer, 2011.isbn: 978 0 85729 020

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Recent progress in the Zimmer program

[Fi19a] D. Fisher. “Groups Acting on Manifolds: Around the Zimmer Program”. In:Group Actions in Ergodic Theory, Geometry, and Topology: Selected Papers. Edited by D. Fisher. Chicago: University of Chicago Press, 2019, pages 609–684. [Fi19b] D. Fisher.Recent progress in the Zimmer program. arXiv:1711.07089 [math]. Feb

work page internal anchor Pith review Pith/arXiv arXiv 2019
[11]

Actions de r´ eseaux sur le cercle

Edited by S.-T. Yau. Surveys in Differential Geometry. In honor of Calabi, Lawson, Siu, and Uhlenbeck. International Press, Somerville, MA, 2003, pages 191–234. [Gh99] ´E. Ghys. “Actions de r´ eseaux sur le cercle”. In:Inventiones Mathematicae 137.1 (1999), pages 199–231. [La08] V. Lafforgue. “Un renforcement de la propri´ et´ e (T)”. In:Duke Mathematical...

2003
[12]

Can lattices in SL(n,R) act on the circle?

[Ma97] I. Maleshich. “The Hilbert-Smith conjecture for H¨ older actions”. In:Uspekhi Matematicheskikh Nauk52.2 (1997). Translated inRussian Mathematical Surveys, 52(2):407–408, 1997, pages 173–174. [Mo09] D. W. Morris.Can lattices in SL(n,R) act on the circle?arXiv:0811.0051 [math]. Feb

work page internal anchor Pith review Pith/arXiv arXiv 1997
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[Mo24] D. W. Morris.Non-left-orderability of lattices in higher-rank semisimple Lie groups (after Deroin and Hurtado). arXiv:2407.09742 [math]. July

work page arXiv
[14]

Invariant measures for actions of unipotent groups over local fields on homogeneous spaces

[MT94] G. A. Margulis and G. M. Tomanov. “Invariant measures for actions of unipotent groups over local fields on homogeneous spaces”. en. In:Inventiones mathematicae116.1 (Dec. 1994), pages 347–392.issn: 1432-1297. [MZ55] D. Montgomery and L. Zippin.Topological Transformation Groups. Volume

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Groups of Circle Diffeomorphisms

[Na02] A. Navas. “Actions de groupes de Kazhdan sur le cercle”. In:Annales Scientifiques de l’ ´Ecole Normale Sup´ erieure (4)35.5 (2002), pages 749–758. [Na09] A. Navas.Groups of Circle Diffeomorphisms. arXiv:math/0607481. May

work page internal anchor Pith review Pith/arXiv arXiv 2002
[16]

A theorem on periodic transformations of spaces

[Ne31] M. H. A. Newman. “A theorem on periodic transformations of spaces”. In: Quarterly Journal of Mathematics2 (1931), pages 1–8. [Pa13] J. Pardon. “The Hilbert–Smith conjecture for three-manifolds”. In:Journal of the American Mathematical Society26.3 (2013), pages 879–899. [Ra90a] M. Ratner. “On measure rigidity of unipotent subgroups of semisimple gro...

1931
[17]

Low dimensional actions of semisimple groups

[St92] G. Stuck. “Low dimensional actions of semisimple groups”. In:Israel Journal of Mathematics80.1 (June 1992), pages 255–255.issn: 1565-8511. [Ti72] J. Tits. “Free subgroups in linear groups”. In:Journal of Algebra20.2 (Feb. 1972), pages 250–270.issn: 0021-8693. [Wi94] D. Witte. “Arithmetic groups of higher Q-rank cannot act on 1-manifolds”. In:Procee...

1992
[18]

Spectrum, entropy, and geometric structures for smooth actions of kazhdan groups

Providence, RI: American Mathematical Society, 1986, pages 1247–1258. [Zi91] R. J. Zimmer. “Spectrum, entropy, and geometric structures for smooth actions of kazhdan groups”. en. In:Israel Journal of Mathematics75.1 (Feb. 1991), pages 65–80.issn: 1565-8511. Department of Mathematics, ETH Z ¨urich, Z ¨urich, Switzerland Email address:segev.gonencohen@math.ethz.ch

1986

[1] [1]

Extremal Lyapunov exponents: an invariance principle and applications

[AV10] A. Avila and M. Viana. “Extremal Lyapunov exponents: an invariance principle and applications”. en. In:Inventiones mathematicae181.1 (July 2010), pages 115–178. issn: 1432-1297. [BDZ22] A. Brown, D. Damjanovi´ c, and Z. Zhang. “C1 actions on manifolds by lattices in Lie groups”. en. In:Compositio Mathematica158.3 (Mar. 2022), pages 529–549.issn: 00...

work page arXiv 2010

[2] [2]

Zimmer’s conjecture: Subexponential growth, measure rigidity, and strong property (T)

[BFH22] A. Brown, D. Fisher, and S. Hurtado. “Zimmer’s conjecture: Subexponential growth, measure rigidity, and strong property (T)”. In:Annals of Mathematics 196.3 (2022), pages 891–940. [BHV08] B. Bekka, P. de la Harpe, and A. Valette.Kazhdan’s Property (T). Volume

2022

[3] [3]

Smooth ergodic theory ofZ d-actions

[BHW23] A. Brown, F. R. Hertz, and Z. Wang. “Smooth ergodic theory ofZ d-actions”. en. In:Journal of Modern Dynamics19.0 (May 2023). Publisher: Journal of Modern Dynamics, pages 455–540.issn: 1930-5311. [BM02] M. Burger and N. Monod. “Continuous bounded cohomology and applications to rigidity theory”. In:Geom. Funct. Anal.12.2 (2002), pages 219–280. [BM46...

2023

[4] [4]

New York: Springer-Verlag, 1991.isbn: 978-0-387-97370-8

Graduate Texts in Mathematics. New York: Springer-Verlag, 1991.isbn: 978-0-387-97370-8. 26 [BP07] L. Barreira and Y. Pesin.Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents. Volume

1991

[5] [5]

Entropy, Lyapunov exponents, and rigidity of group actions

[Br19] A. Brown. “Entropy, Lyapunov exponents, and rigidity of group actions”. en. In:Ensaios Matem´ aticos33.2019 (2019).issn: 21750432. [Br72] G. E. Bredon.Introduction to Compact Transformation Groups. Volume

2019

[6] [6]

Progr` es r´ ecents concernant le programme de Zimmer (d’apr` es A. Brown, D. Fisher & S. Hurtado)

Pure and Applied Mathematics. New York and London: Academic Press, 1972.isbn: 978-0-12-128850-1. [Ca17] S. Cantat. “Progr` es r´ ecents concernant le programme de Zimmer (d’apr` es A. Brown, D. Fisher & S. Hurtado)”. In:S´ eminaire Bourbaki. 70 e ann´ ee, 2017–2018, no. 1136(2017), page

1972

[7] [7]

Algebraic actions of discrete groups: thep-adic method

[CX18] S. Cantat and J. Xie. “Algebraic actions of discrete groups: thep-adic method”. In:Acta Mathematica220.2 (2018), pages 239–295. [DH20] B. Deroin and S. Hurtado.Non left-orderability of lattices in higher rank semi-simple Lie groups. arXiv:2008.10687 [math]. Aug

work page arXiv 2018

[8] [8]

Diagonal Actions on Locally Homogeneous Spaces

[EL10] M. Einsiedler and E. Lindenstrauss. “Diagonal Actions on Locally Homogeneous Spaces”. In:Homogeneous flows, moduli spaces, and arithmetic. proceedings of the Clay Mathematics Institute Summer School. American Mathematical Society, 2010, pages 155–242. [EW11] M. Einsiedler and T. Ward.Ergodic Theory with a View Towards Number Theory. Volume

2010

[9] [9]

Springer, 2011.isbn: 978 0 85729 020

Graduate Texts in Mathematics. Springer, 2011.isbn: 978 0 85729 020

2011

[10] [10]

Recent progress in the Zimmer program

[Fi19a] D. Fisher. “Groups Acting on Manifolds: Around the Zimmer Program”. In:Group Actions in Ergodic Theory, Geometry, and Topology: Selected Papers. Edited by D. Fisher. Chicago: University of Chicago Press, 2019, pages 609–684. [Fi19b] D. Fisher.Recent progress in the Zimmer program. arXiv:1711.07089 [math]. Feb

work page internal anchor Pith review Pith/arXiv arXiv 2019

[11] [11]

Actions de r´ eseaux sur le cercle

Edited by S.-T. Yau. Surveys in Differential Geometry. In honor of Calabi, Lawson, Siu, and Uhlenbeck. International Press, Somerville, MA, 2003, pages 191–234. [Gh99] ´E. Ghys. “Actions de r´ eseaux sur le cercle”. In:Inventiones Mathematicae 137.1 (1999), pages 199–231. [La08] V. Lafforgue. “Un renforcement de la propri´ et´ e (T)”. In:Duke Mathematical...

2003

[12] [12]

Can lattices in SL(n,R) act on the circle?

[Ma97] I. Maleshich. “The Hilbert-Smith conjecture for H¨ older actions”. In:Uspekhi Matematicheskikh Nauk52.2 (1997). Translated inRussian Mathematical Surveys, 52(2):407–408, 1997, pages 173–174. [Mo09] D. W. Morris.Can lattices in SL(n,R) act on the circle?arXiv:0811.0051 [math]. Feb

work page internal anchor Pith review Pith/arXiv arXiv 1997

[13] [13]

[Mo24] D. W. Morris.Non-left-orderability of lattices in higher-rank semisimple Lie groups (after Deroin and Hurtado). arXiv:2407.09742 [math]. July

work page arXiv

[14] [14]

Invariant measures for actions of unipotent groups over local fields on homogeneous spaces

[MT94] G. A. Margulis and G. M. Tomanov. “Invariant measures for actions of unipotent groups over local fields on homogeneous spaces”. en. In:Inventiones mathematicae116.1 (Dec. 1994), pages 347–392.issn: 1432-1297. [MZ55] D. Montgomery and L. Zippin.Topological Transformation Groups. Volume

1994

[15] [15]

Groups of Circle Diffeomorphisms

[Na02] A. Navas. “Actions de groupes de Kazhdan sur le cercle”. In:Annales Scientifiques de l’ ´Ecole Normale Sup´ erieure (4)35.5 (2002), pages 749–758. [Na09] A. Navas.Groups of Circle Diffeomorphisms. arXiv:math/0607481. May

work page internal anchor Pith review Pith/arXiv arXiv 2002

[16] [16]

A theorem on periodic transformations of spaces

[Ne31] M. H. A. Newman. “A theorem on periodic transformations of spaces”. In: Quarterly Journal of Mathematics2 (1931), pages 1–8. [Pa13] J. Pardon. “The Hilbert–Smith conjecture for three-manifolds”. In:Journal of the American Mathematical Society26.3 (2013), pages 879–899. [Ra90a] M. Ratner. “On measure rigidity of unipotent subgroups of semisimple gro...

1931

[17] [17]

Low dimensional actions of semisimple groups

[St92] G. Stuck. “Low dimensional actions of semisimple groups”. In:Israel Journal of Mathematics80.1 (June 1992), pages 255–255.issn: 1565-8511. [Ti72] J. Tits. “Free subgroups in linear groups”. In:Journal of Algebra20.2 (Feb. 1972), pages 250–270.issn: 0021-8693. [Wi94] D. Witte. “Arithmetic groups of higher Q-rank cannot act on 1-manifolds”. In:Procee...

1992

[18] [18]

Spectrum, entropy, and geometric structures for smooth actions of kazhdan groups

Providence, RI: American Mathematical Society, 1986, pages 1247–1258. [Zi91] R. J. Zimmer. “Spectrum, entropy, and geometric structures for smooth actions of kazhdan groups”. en. In:Israel Journal of Mathematics75.1 (Feb. 1991), pages 65–80.issn: 1565-8511. Department of Mathematics, ETH Z ¨urich, Z ¨urich, Switzerland Email address:segev.gonencohen@math.ethz.ch

1986