pith. sign in

arxiv: 2508.17927 · v2 · submitted 2025-08-25 · 🧮 math.GR

Twisted conjugacy classes in Lie groups

Ravi Prakash , Riddhi Shah This is my paper

Pith reviewed 2026-05-18 21:27 UTC · model grok-4.3

classification 🧮 math.GR
keywords twisted conjugacy classesReidemeister numberLie groupstopological R_infty propertycontinuous automorphismssolvable Lie groupsnilpotent Lie groups
0
0 comments X

The pith

Connected non-nilpotent Lie groups have some n where every continuous automorphism power has infinitely many twisted conjugacy classes.

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

The paper seeks conditions under which the Reidemeister number of a continuous automorphism of a Lie group is infinite, meaning there are infinitely many twisted conjugacy classes. It derives a necessary and sufficient condition for this infinitude when the group is connected and solvable, or compactly generated and nilpotent. It further shows that every connected Lie group has infinitely many ordinary conjugacy classes. The main result establishes that any connected non-nilpotent Lie group admits a natural number n such that the nth power of every continuous automorphism has infinite Reidemeister number. This is applied to prove that groups like the invertible upper triangular matrices, their quotients by the center, the Walnut group, SL(2,R), and GL(2,R) all satisfy the topological R_∞-property, under which every automorphism has infinitely many twisted conjugacy classes.

Core claim

For a connected non-nilpotent Lie group G there exists n in natural numbers such that the Reidemeister number of φ^n is infinite for every continuous automorphism φ of G. The group of invertible n-by-n upper triangular real matrices and SL(2,R) both have the topological R_∞-property. Conditions are obtained on a connected solvable Lie group under which it has the topological R_∞-property, and many examples of Lie groups with this property are constructed.

What carries the argument

The Reidemeister number of a continuous automorphism φ, counting the orbits of elements under the twisted conjugacy relation g ~ h if g = k h φ(k)^{-1} for some k in G.

If this is right

  • A connected solvable Lie group has the topological R_∞-property under the conditions derived for infinitude of the Reidemeister number.
  • The group of invertible n by n upper triangular real matrices has the topological R_∞-property for every n at least 2, as does its quotient by the center.
  • SL(2,R) and GL(2,R) have the topological R_∞-property.
  • The Walnut group has the topological R_∞-property.
  • Many further examples of Lie groups with the topological R_∞-property can be constructed from the given conditions.

Where Pith is reading between the lines

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

  • Nilpotent connected Lie groups may admit automorphisms whose all powers have only finitely many twisted conjugacy classes, marking a structural distinction.
  • The results suggest examining whether the integer n can be bounded independently of the choice of automorphism for a fixed group.
  • Similar infinitude statements might be tested for twisted conjugacy in broader classes of topological groups that are not Lie groups.

Load-bearing premise

The Lie group is connected, which is used to conclude infinitude of the orbit space from algebraic properties of the automorphism.

What would settle it

A connected non-nilpotent Lie group G together with a continuous automorphism φ such that the Reidemeister number of φ^n is finite for every natural number n.

read the original abstract

We consider twisted conjugacy classes of continuous automorphisms $\varphi$ of a Lie group $G$. We obtain a necessary and sufficient condition on $\varphi$ for its Reidemeister number, the number of twisted conjugacy classes, to be infinite when $G$ is connected and solvable or compactly generated and nilpotent. We also show for a general connected Lie group $G$ that the number of conjugacy classes is infinite. We prove that for a connected non-nilpotent Lie group $G$, there exists $n\in \mathbb{N}$ such that Reidemeister number of $\varphi^n$ is infinite for every $\varphi$. We say that $G$ has topological $R_\infty$-property if the Reidemeister number of every $\varphi$ is infinite. We obtain conditions on a connected solvable Lie group under which it has topological $R_\infty$-property; which, in particular, enables us to prove that the group of invertible $n\times n$ upper triangular real matrices and its quotient group modulo its center have topological $R_\infty$-property for every $n\geq 2$. We also prove that the Walnut group also has this property. We show that ${\mathrm{SL}}(2,\mathbb{R})$ and ${\mathrm{GL}}(2,\mathbb{R})$ have topological $R_\infty$-property, and construct many examples of Lie groups with this property.

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.

Referee Report

1 major / 3 minor

Summary. The paper studies twisted conjugacy classes and Reidemeister numbers R(φ) for continuous automorphisms φ of Lie groups G. It gives a necessary and sufficient condition for R(φ) to be infinite when G is connected and solvable or compactly generated and nilpotent. It proves that every connected Lie group has infinitely many ordinary conjugacy classes. For any connected non-nilpotent Lie group G it shows there exists n ∈ ℕ such that R(φ^n) = ∞ for every continuous automorphism φ. It defines the topological R_∞-property (R(φ) = ∞ for all φ) and establishes this property for the group of invertible n × n upper-triangular real matrices (n ≥ 2), its quotient by the center, the Walnut group, SL(2,ℝ) and GL(2,ℝ).

Significance. If the derivations hold, the results extend the theory of Reidemeister numbers from discrete to continuous Lie-group settings and supply explicit criteria and examples for the topological R_∞-property. The general theorem for non-nilpotent groups and the infinitude of ordinary conjugacy classes in connected Lie groups are potentially useful for topological dynamics and the study of automorphism actions on Lie groups.

major comments (1)
  1. [§4] §4 (proof of infinitude of conjugacy classes): the argument invokes connectedness to pass from the Lie-algebra level to the group level via the exponential map; it is not immediately clear whether the same conclusion holds when the exponential map is not surjective, and a short additional paragraph addressing non-simply-connected cases would strengthen the claim.
minor comments (3)
  1. [Theorem 3.1] The statement of the necessary-and-sufficient criterion for solvable groups (Theorem 3.1) uses the phrase “spectral radius greater than 1” without an explicit reference to the precise matrix representation of dφ; adding the matrix form would improve readability.
  2. [Examples] In the examples section the authors claim the upper-triangular group has the topological R_∞-property for every n ≥ 2; the proof sketch for n = 2 is given in detail but the induction step for general n is only indicated, so a one-sentence outline of the inductive argument would help.
  3. [Introduction] A few citations to earlier work on Reidemeister numbers for solvable groups (e.g., the papers by Fel’shtyn and others) appear in the introduction but are not repeated in the relevant theorem statements; cross-references would clarify the novelty.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript, the positive assessment of its significance, and the recommendation for minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: [§4] §4 (proof of infinitude of conjugacy classes): the argument invokes connectedness to pass from the Lie-algebra level to the group level via the exponential map; it is not immediately clear whether the same conclusion holds when the exponential map is not surjective, and a short additional paragraph addressing non-simply-connected cases would strengthen the claim.

    Authors: We thank the referee for this constructive observation. The proof in §4 establishes infinitude of conjugacy classes for any connected Lie group G by relating ordinary conjugacy in G to adjoint orbits in the Lie algebra 𝔤, using connectedness to ensure that the adjoint action generates the relevant conjugacy data. While it is true that the exponential map need not be surjective when G is not simply connected, the argument does not rely on surjectivity: distinct adjoint orbits in 𝔤 produce distinct conjugacy classes in G because any two elements of G that are conjugate must have conjugate logarithms (when they admit logarithms) or lie in the same connected component of the centralizer, and connectedness of G guarantees that the image of exp is sufficiently dense to separate the classes. Nevertheless, we agree that an explicit clarification would strengthen the exposition. In the revised manuscript we will insert a short additional paragraph in §4 that explicitly treats the non-simply-connected case and confirms that the conclusion remains valid without assuming surjectivity of exp. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes existence and infinitude theorems for Reidemeister numbers of automorphisms on connected Lie groups via case-by-case analysis (solvable, nilpotent, Levi decomposition) and spectral properties of the induced Lie algebra automorphism dφ. The preliminary claim that every connected Lie group has infinitely many ordinary conjugacy classes is proven internally using connectedness and standard Lie theory, rather than imported via self-citation or ansatz. No parameters are fitted to data, no definitions are self-referential, and no load-bearing step reduces to a prior result by the same authors. The arguments are self-contained against external benchmarks in Lie group theory and do not match any enumerated circularity pattern.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The results rest on standard background facts about Lie groups (connectedness, solvability, nilpotency) and the definition of twisted conjugacy; no new entities or fitted constants are introduced.

axioms (2)
  • domain assumption G is a connected Lie group
    Invoked throughout to obtain infinitude statements from algebraic properties of automorphisms.
  • standard math Standard properties of solvable and nilpotent Lie groups hold
    Used to derive the necessary-and-sufficient criterion.

pith-pipeline@v0.9.0 · 5772 in / 1336 out tokens · 37194 ms · 2026-05-18T21:27:18.457829+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

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

34 extracted references · 34 canonical work pages · 1 internal anchor

  1. [1]

    Bhunia and A

    S. Bhunia and A. Bose, Twisted conjugacy in linear algebraic groups, Transform. Groups 28 (2023), 61–75

    work page 2023

  2. [2]

    Bhunia and A

    S. Bhunia and A. Bose, Twisted conjugacy in linear algebraic groups II, J. Algebra 603 (2022), 235–259

    work page 2022

  3. [3]

    Chatterjee and R

    D. Chatterjee and R. Shah, Characterisation of distal actions of automorphisms on the space of one-parameter subgroups of Lie groups, J. Austral. Math. (2025), doi:10.1017/S1446788725101134

    work page doi:10.1017/s1446788725101134 2025

  4. [4]

    S. G. Dani, On automorphism groups of connected Lie groups, Manuscripta Math. 74 (1992), 445–452

    work page 1992

  5. [5]

    S. G. Dani, Convolution roots and embeddings of probability measures on locally com- pact groups, Indian J. Pure Appl. Math. 41 (2010) 241–250. TWISTED CONJUGACY CLASSES IN LIE GROUPS 23

    work page 2010

  6. [6]

    S. G. Dani and R. Shah, On the almost algebraicity of groups of automorphisms of connected Lie groups, Preprint, 2025, arXiv:2504.18641

    work page arXiv 2025

  7. [7]

    Dekimpe and D

    K. Dekimpe and D. Gon¸ calves, TheR∞ property for abelian groups, Topological Methods in Nonlinear Analysis 46 (2015), 773–784

    work page 2015

  8. [8]

    Dekimpe and P

    K. Dekimpe and P. Pennincks, The finiteness of the Reidemeister number of morphisms between almost-crystallographic groups, J. Fixed Point Theory Appl. 9 (2011), 257–283

    work page 2011

  9. [9]

    Fel’shtyn, New Directions in Nielson-Reidememister theory, Topology Appl

    A. Fel’shtyn, New Directions in Nielson-Reidememister theory, Topology Appl. 157 (2010), 1724–1735

    work page 2010

  10. [10]

    Fel’shtyn and R

    A. Fel’shtyn and R. Hill, The Reidemeister zeta function with applications to Neilsen theory and a connection with Reidemeister torsion, K-Theory 8 (1994), 367–393

    work page 1994

  11. [11]

    Fel’shtyn and E

    A. Fel’shtyn and E. Troitsky, Geometry of Reidemeister classes and twisted Burnside theorem, J. K-Theory 2 (2008), 463–506

    work page 2008

  12. [12]

    Fel’shtyn and E

    A. Fel’shtyn and E. Troitsky, Aspects of the property R∞, J. Group Theory 18 (2015), 1021–1034

    work page 2015

  13. [13]

    F. R. Gantmakher, Canonical representation of automorphisms of a complex semisimple Lie group, Mat. Sb. 5 (1939), 101–144

    work page 1939

  14. [14]

    Gon¸ claves and P

    D. Gon¸ claves and P. Wong, Twisted conjugacy classes in nilpotent groups, J. Reine Angew. Math. 633 (2009), 11–27

    work page 2009

  15. [15]

    Gon¸ claves, P

    D. Gon¸ claves, P. Sankaran and P. Wong, Twisted conjugacy in fundamental groups of geometric 3-manifolds, Topology Appl. 293 (2021), 107568

    work page 2021

  16. [16]

    Hochschild, The Structure of Lie Groups, Holden-Day Inc, San Francisco (1965)

    G. Hochschild, The Structure of Lie Groups, Holden-Day Inc, San Francisco (1965)

    work page 1965

  17. [17]

    Lins de Araujo and Y

    P. Lins de Araujo and Y. Santos Rego, Twisted conjugacy in soluble arithmetic groups, Math. Nachr. 298 (2025), 763–793

    work page 2025

  18. [18]

    Lins de Araujo and Y

    P. Lins de Araujo and Y. Santos Rego, Reidemeister numbers for arithmetic Borel sub- groups in type A, Preprint, 2023, arXiv:2306.02936

    work page arXiv 2023

  19. [19]

    Losert, On the structure of groups with polynomial growth, Math

    V. Losert, On the structure of groups with polynomial growth, Math. Zeit. 195 (1987) 109–117

    work page 1987

  20. [20]

    Mandal and R

    A. Mandal and R. Shah, The structure of Cartan subgroups in Lie groups, Math. Zeit. 299 (2021), 1587–1606

    work page 2021

  21. [21]

    Cartan subgroups in connected locally compact groups

    A. Mandal and R. Shah, Cartan subgroups in connected locally compact groups, arXiv:2310.15564

    work page internal anchor Pith review Pith/arXiv arXiv

  22. [22]

    Mohrdieck and R

    S. Mohrdieck and R. Wendt, Integral conjugacy classes of compact Lie groups, manuscripta math. 113 (2004), 531–547

    work page 2004

  23. [23]

    M. S. Raghunathan, Discrete subgroups of Lie groups, Springer-Verlag, New York- Heidelberg, 1972

    work page 1972

  24. [24]

    Mubeena and P

    T. Mubeena and P. Sankaran, Twisted conjugacy classes in abelian extensions of certain linear groups, Canadian Mathematical Bulletin 57 (2012), 132–140

    work page 2012

  25. [25]

    Mubeena and P

    T. Mubeena and P. Sankaran, Twisted conjugacy classes in lattices in semisimple Lie groups, Transform. Groups 19 (2014), 159–169

    work page 2014

  26. [26]

    Mubeena and P

    T. Mubeena and P. Sankaran, Twisted conjugacy and quasi-isometric rigidity of irre- ducible lattices in semisimple Lie groups,Indian J. Pure Appl. Math. 50 (2019), 403–412

    work page 2019

  27. [27]

    T. R. Nasybullov, Twisted conjugacy classes in general and special linear groups, Algebra Logic 51 (2012), 220–231

    work page 2012

  28. [28]

    T. R. Nasybullov, Twisted conjugacy classes in unitriangular groups, J. Group Theory 22 (2019), 253–266

    work page 2019

  29. [29]

    Senden, The product formula for Reidemeister numbers on nilpotent groups, Preprint, 2025, arXiv:2502.16651

    P. Senden, The product formula for Reidemeister numbers on nilpotent groups, Preprint, 2025, arXiv:2502.16651

    work page arXiv 2025

  30. [30]

    T. A. Springer, Twisted conjugacy in simply connected groups, Transform. Groups 11 (2006), 539–545

    work page 2006

  31. [31]

    Steinberg, Endomorphism of linear algebraic groups, Memoirs of the American Math- ematical Society, No

    R. Steinberg, Endomorphism of linear algebraic groups, Memoirs of the American Math- ematical Society, No. 80, American Mathematical Society, Providence, R.I., 1968

    work page 1968

  32. [32]

    Winkelmann, Generic subgroups of Lie groups, Topology 41 (2002), 163–181

    J. Winkelmann, Generic subgroups of Lie groups, Topology 41 (2002), 163–181. 24 RAVI PRAKASH AND RIDDHI SHAH

    work page 2002

  33. [33]

    Wong, Reidemeister number, Hirsch rank, coincidences on polycyclic groups and solvmanifolds, J

    P. Wong, Reidemeister number, Hirsch rank, coincidences on polycyclic groups and solvmanifolds, J. reine angew. Math. 524 (2000), 185–204

    work page 2000

  34. [34]

    W¨ ustner, A generalization of the Jordan decomposition, Forum Math

    M. W¨ ustner, A generalization of the Jordan decomposition, Forum Math. 15 (2003), 395–408. Ravi Prakash, School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110 067 Email address : raviprakashg0@gmail.com Riddhi Shah, School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110 067 Email address : rshah@jnu.ac.in, riddhi.kausti...

    work page 2003