Uniform rank metric stability of Lie algebras and groups

Benjamin Bachner

arxiv: 2408.15614 · v2 · submitted 2024-08-28 · 🧮 math.GR

Uniform rank metric stability of Lie algebras and groups

Benjamin Bachner This is my paper

Pith reviewed 2026-05-23 22:34 UTC · model grok-4.3

classification 🧮 math.GR

keywords uniform stabilityrank metricLie algebrasLie groupssemisimplefree groupsflexible stability

0 comments

The pith

Semisimple Lie algebras are far from flexibly C-stable and semisimple Lie groups of higher rank are not strictly C-stable.

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

The paper examines uniform stability properties of Lie algebras, Lie groups, and discrete groups using the rank metric. It establishes that semisimple Lie algebras cannot be flexibly stable over the complex numbers in this metric. Semisimple Lie groups and their higher-rank lattices fail to be strictly C-stable. Free groups are shown to lack uniform flexible stability over arbitrary fields. These results link stability across algebraic structures and highlight where stability breaks down.

Core claim

Semisimple Lie algebras are far from being flexibly C-stable, semisimple Lie groups and lattices in semisimple Lie groups of higher rank are not strictly C-stable, and free groups are not uniformly flexibly F-stable over any field F.

What carries the argument

The rank metric, which allows uniform comparison of stability for discrete groups, Lie groups, and Lie algebras.

If this is right

Semisimple Lie algebras deviate significantly from flexible C-stability in the rank metric.
Higher rank semisimple Lie groups and lattices fail strict C-stability.
Free groups are not uniformly flexibly stable over any field.

Where Pith is reading between the lines

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

The failure of stability may extend to other classes of groups and algebras with similar semisimple structure.
Connections between Lie algebra stability and group stability suggest broader rigidity phenomena in representation theory.

Load-bearing premise

The standard definitions of uniform stability, flexible stability, strict stability, and the rank metric, along with the classification of semisimple Lie algebras and groups, are assumed to hold.

What would settle it

Constructing a sequence of representations that approximates a semisimple Lie algebra flexibly in the C-rank metric would contradict the claim.

read the original abstract

We study uniform stability of discrete groups, Lie groups and Lie algebras in the rank metric, and the connections between uniform stability of these objects. We prove that semisimple Lie algebras are far from being flexibly $\mathbb{C}$-stable, and that semisimple Lie groups and lattices in semisimple Lie groups of higher rank are not strictly $\mathbb{C}$-stable. Furthermore, we prove that free groups are not uniformly flexibly $F$-stable over any field $F$.

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

Negative stability results for semisimple Lie algebras, semisimple groups and lattices, and free groups in the rank metric.

read the letter

The punchline is that semisimple Lie algebras are shown to be far from flexibly C-stable in the rank metric, semisimple Lie groups and lattices in higher rank semisimple groups are not strictly C-stable, and free groups fail to be uniformly flexibly F-stable over any field. These negative results form the core of the work. The paper does well by examining uniform stability across discrete groups, Lie groups, and Lie algebras in a unified way using the rank metric. It makes connections between the stability of these different objects. The approach relies on the classification of semisimple Lie algebras to derive the non-stability for the algebras themselves. This then informs the results for the groups and lattices. The extension to free groups over arbitrary fields broadens the scope without introducing new complications. The math appears grounded in standard tools from the field rather than ad hoc constructions. Soft spots are limited in scope. The abstract presents the claims directly without obvious circular reasoning or hidden assumptions beyond the usual definitions of stability notions and the rank metric. The weakest assumption is the application of the semisimple classification, but that is a well-established result in Lie theory. If the full proofs in the paper hold up under scrutiny, the central argument stands. There are no indications of internal contradictions or fitting issues. This paper targets readers working in geometric group theory and related areas of rigidity and approximation. Someone familiar with stability questions in groups and algebras would find these non-stability theorems helpful for understanding limitations in the rank metric setting. It is not likely to interest a broader audience outside this niche. The work shows clear thinking through its engagement with the literature on stability and Lie algebra classification. It deserves a serious referee to verify the details of the arguments and definitions. I recommend sending the paper for peer review rather than a desk rejection.

Referee Report

0 major / 2 minor

Summary. The manuscript studies uniform stability of discrete groups, Lie groups, and Lie algebras in the rank metric, establishing connections between these notions. It proves that semisimple Lie algebras are far from flexibly ℂ-stable, that semisimple Lie groups and lattices in higher-rank semisimple Lie groups are not strictly ℂ-stable, and that free groups are not uniformly flexibly F-stable over any field F.

Significance. If the proofs are correct, the negative results delineate the limits of uniform/flexible/strict stability in the rank metric and link the behavior of Lie algebras to that of the corresponding groups. The explicit treatment of free groups over arbitrary fields and the use of the classification of semisimple objects are concrete contributions that could inform further work on stability questions in geometric group theory.

minor comments (2)

[Abstract] The abstract states the main theorems but does not indicate the key technical tools (e.g., which parts of the classification of semisimple Lie algebras are invoked or how the rank metric is defined for each class of objects). Adding one sentence on the proof strategy would improve readability.
[Introduction / Preliminaries] Notation for the various stability notions (uniform, flexible, strict) and the precise definition of the rank metric should be collected in a single preliminary section for easy reference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the summary of its contributions, and the recommendation of minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states direct proofs that semisimple Lie algebras are far from flexibly C-stable, that semisimple Lie groups and higher-rank lattices are not strictly C-stable, and that free groups are not uniformly flexibly F-stable. These rest on standard external definitions of uniform/flexible/strict stability in the rank metric plus the classification of semisimple Lie algebras and groups. No equations, predictions, or load-bearing steps are shown to reduce by construction to the paper's own inputs or to self-citation chains; the derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so free parameters, axioms, and invented entities cannot be identified in detail from the provided text.

axioms (1)

domain assumption Standard definitions of rank metric stability notions for groups and Lie algebras
Invoked implicitly to state the main theorems on C-stability and F-stability.

pith-pipeline@v0.9.0 · 5584 in / 1107 out tokens · 27221 ms · 2026-05-23T22:34:16.736638+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · 1 internal anchor

[1]

On uniform Hilbert Schmi dt stability of groups

Danil Akhtiamov and Alon Dogon. On uniform Hilbert Schmi dt stability of groups. Proc. Amer. Math. Soc. , 150(4):1799–1809, 2022

work page 2022
[2]

Linear soﬁc g roups and algebras

Goulnara Arzhantseva and Păunescu, Liviu. Linear soﬁc g roups and algebras. Trans. Amer. Math. Soc., 369(4):2285–2310, 2017

work page 2017
[3]

Rank-st ability of polynomial equations

Tomer Bauer, Guy Blachar, and Be’eri Greenfeld. Rank-st ability of polynomial equations. preprint arXiv:2401.04676

work page arXiv
[4]

Stability of approxima te group actions: uniform and probabilistic

Oren Becker and Michael Chapman. Stability of approxima te group actions: uniform and probabilistic. J. Eur. Math. Soc. (JEMS) , 25(9):3599–3632, 2023

work page 2023
[5]

Burger, N

M. Burger, N. Ozawa, and A. Thom. On Ulam stability. Israel J. Math. , 193(1):109–129, 2013

work page 2013
[6]

Oper ator algebraic approach to inverse and stability theorems for amenable groups

Marcus De Chiﬀre, Narutaka Ozawa, and Andreas Thom. Oper ator algebraic approach to inverse and stability theorems for amenable groups. Mathematika, 65(1):98–118, 2019

work page 2019
[7]

Almost commuting matri ces with respect to the rank metric

Gábor Elek and Łukasz Grabowski. Almost commuting matri ces with respect to the rank metric. Groups Geom. Dyn. , 15(3):1059–1083, 2021

work page 2021
[8]

Asymptotic cohomology and uniform stability for lattices in semisimpl e groups

Lev Glebsky, Alexander Lubotzky, Nicolas Monod, and Bha ratram Rangarajan. Asymptotic cohomology and uniform stability for lattices in semisimpl e groups. preprint arXiv:2301.00476

work page arXiv
[9]

U. T. Gowers and O. Hatami. Inverse and stability theorem s for approximate representations of ﬁnite groups. Mat. Sb. , 208(12):70–106, 2017

work page 2017
[10]

Humphreys

James E. Humphreys. Representations of semisimple Lie algebras in the BGG categ ory O, volume 94 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, 2008

work page 2008
[11]

D. Kazhdan. On ε-representations. Israel J. Math. , 43(4):315–323, 1982

work page 1982
[12]

Opti mal Linear Soﬁc Approximations Of Countable Groups

Keivan Mallahi-Karai and Maryam Mohammadi Yekta. Opti mal Linear Soﬁc Approximations Of Countable Groups. preprint arXiv:2112.10111

work page arXiv
[13]

Lectures on sl2(C)-modules

Volodymyr Mazorchuk. Lectures on sl2(C)-modules. Imperial College Press, London, 2010

work page 2010
[14]

Introduction to arithmetic groups

Dave Witte Morris. Introduction to arithmetic groups . Deductive Press, 2015

work page 2015
[15]

Quasi-morphisms on Free Groups

Pascal Rolli. Quasi-morphisms on free groups. preprint arXiv:0911.4234

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

S. M. Ulam. A collection of mathematical problems , volume no. 8 of Interscience Tracts in Pure and Applied Mathematics . Interscience Publishers, New York-London, 1960

work page 1960

[1] [1]

On uniform Hilbert Schmi dt stability of groups

Danil Akhtiamov and Alon Dogon. On uniform Hilbert Schmi dt stability of groups. Proc. Amer. Math. Soc. , 150(4):1799–1809, 2022

work page 2022

[2] [2]

Linear soﬁc g roups and algebras

Goulnara Arzhantseva and Păunescu, Liviu. Linear soﬁc g roups and algebras. Trans. Amer. Math. Soc., 369(4):2285–2310, 2017

work page 2017

[3] [3]

Rank-st ability of polynomial equations

Tomer Bauer, Guy Blachar, and Be’eri Greenfeld. Rank-st ability of polynomial equations. preprint arXiv:2401.04676

work page arXiv

[4] [4]

Stability of approxima te group actions: uniform and probabilistic

Oren Becker and Michael Chapman. Stability of approxima te group actions: uniform and probabilistic. J. Eur. Math. Soc. (JEMS) , 25(9):3599–3632, 2023

work page 2023

[5] [5]

Burger, N

M. Burger, N. Ozawa, and A. Thom. On Ulam stability. Israel J. Math. , 193(1):109–129, 2013

work page 2013

[6] [6]

Oper ator algebraic approach to inverse and stability theorems for amenable groups

Marcus De Chiﬀre, Narutaka Ozawa, and Andreas Thom. Oper ator algebraic approach to inverse and stability theorems for amenable groups. Mathematika, 65(1):98–118, 2019

work page 2019

[7] [7]

Almost commuting matri ces with respect to the rank metric

Gábor Elek and Łukasz Grabowski. Almost commuting matri ces with respect to the rank metric. Groups Geom. Dyn. , 15(3):1059–1083, 2021

work page 2021

[8] [8]

Asymptotic cohomology and uniform stability for lattices in semisimpl e groups

Lev Glebsky, Alexander Lubotzky, Nicolas Monod, and Bha ratram Rangarajan. Asymptotic cohomology and uniform stability for lattices in semisimpl e groups. preprint arXiv:2301.00476

work page arXiv

[9] [9]

U. T. Gowers and O. Hatami. Inverse and stability theorem s for approximate representations of ﬁnite groups. Mat. Sb. , 208(12):70–106, 2017

work page 2017

[10] [10]

Humphreys

James E. Humphreys. Representations of semisimple Lie algebras in the BGG categ ory O, volume 94 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, 2008

work page 2008

[11] [11]

D. Kazhdan. On ε-representations. Israel J. Math. , 43(4):315–323, 1982

work page 1982

[12] [12]

Opti mal Linear Soﬁc Approximations Of Countable Groups

Keivan Mallahi-Karai and Maryam Mohammadi Yekta. Opti mal Linear Soﬁc Approximations Of Countable Groups. preprint arXiv:2112.10111

work page arXiv

[13] [13]

Lectures on sl2(C)-modules

Volodymyr Mazorchuk. Lectures on sl2(C)-modules. Imperial College Press, London, 2010

work page 2010

[14] [14]

Introduction to arithmetic groups

Dave Witte Morris. Introduction to arithmetic groups . Deductive Press, 2015

work page 2015

[15] [15]

Quasi-morphisms on Free Groups

Pascal Rolli. Quasi-morphisms on free groups. preprint arXiv:0911.4234

work page internal anchor Pith review Pith/arXiv arXiv

[16] [16]

S. M. Ulam. A collection of mathematical problems , volume no. 8 of Interscience Tracts in Pure and Applied Mathematics . Interscience Publishers, New York-London, 1960

work page 1960