Homomorphisms from topological groups to inverse limits

Curtis Kent; Gregory R. Conner; Samuel M. Corson

arxiv: 2607.01411 · v1 · pith:BNBQBMXCnew · submitted 2026-07-01 · 🧮 math.GR · math.LO

Homomorphisms from topological groups to inverse limits

Gregory R. Conner , Samuel M. Corson , Curtis Kent This is my paper

Pith reviewed 2026-07-03 01:23 UTC · model grok-4.3

classification 🧮 math.GR math.LO

keywords topological groupsPolish groupshomomorphismsinverse limitsautomatic continuityultraproductsGrigorchuk groupresidually finite groups

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The pith

Every homomorphism from a Polish group to a countable torsion-free residually finite group has open kernel.

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

The paper establishes a general theorem that imposes constraints on homomorphisms from certain topological groups into inverse limits of bounded torsion groups. This theorem implies automatic continuity results, such as that any homomorphism from a Polish group to a countable torsion-free residually finite group must have an open kernel. It also yields a result on ultraproducts: the Grigorchuk group appears as a homomorphic image of a nonprincipal ultraproduct of groups precisely when a measurable cardinal exists. These outcomes connect topological properties of groups with algebraic and set-theoretic conditions.

Core claim

A general theorem constrains homomorphisms from specified topological groups to inverse limits of bounded torsion groups, from which it follows that homomorphisms from Polish groups to countable torsion-free residually finite groups have open kernels, and that the Grigorchuk group is a homomorphic image of a nonprincipal ultraproduct if and only if a measurable cardinal exists.

What carries the argument

The general theorem providing constraints on maps from certain topological groups to inverse limits of bounded torsion groups.

If this is right

Any homomorphism from a Polish group to a countable torsion-free residually finite group has an open kernel.
The Grigorchuk group is a homomorphic image of a nonprincipal ultraproduct of groups exactly when there exists a measurable cardinal.
Automatic continuity holds for certain classes of homomorphisms between topological groups.

Where Pith is reading between the lines

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

Such results may limit the possible continuous images of Polish groups in the category of residually finite groups.
Existence of measurable cardinals becomes necessary for certain algebraic constructions involving the Grigorchuk group.
Similar constraints might apply to other classes of topological groups beyond Polish ones if the domain conditions can be relaxed.

Load-bearing premise

The domain groups in the general theorem satisfy the required topological properties that allow the constraints on homomorphisms to inverse limits to hold.

What would settle it

Constructing a homomorphism from a Polish group to a countable torsion-free residually finite group whose kernel is not open would disprove the automatic continuity claim.

read the original abstract

We prove a general theorem giving constraints on maps from certain topological groups to inverse limits of bounded torsion groups. From this we obtain some automatic continuity and ultraproduct results. For example, every homomorphism from a Polish group to a countable torsion-free residually finite group has open kernel. Also, the Grigorchuk group is a homomorphic image of a nonprincipal ultraproduct of groups if and only if there exists a measurable cardinal.

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

The paper's general theorem on homomorphisms to inverse limits of bounded-torsion groups directly yields an automatic continuity result for Polish groups and an equivalence tying the Grigorchuk group to measurable cardinals via ultraproducts.

read the letter

The main point to know is that the authors prove a general theorem constraining homomorphisms from certain topological groups to inverse limits of bounded-torsion groups. From this they derive that homs out of Polish groups into countable torsion-free residually finite groups have open kernels, and that the Grigorchuk group arises as a homomorphic image of a nonprincipal ultraproduct precisely when there is a measurable cardinal.

The general theorem is the novel contribution here. It seems to offer a single argument that covers both an automatic continuity result and a set-theoretic characterization involving ultraproducts. The Polish group application is a solid, concrete statement that builds on known techniques in the area. The Grigorchuk result is precise and ties the group directly to the existence of large cardinals.

What the paper does well is present the general result upfront and then show how the two examples follow as direct consequences. This makes the logic transparent. The claims match the flavor of existing work on automatic continuity in Polish groups and on ultraproducts in group theory.

The soft spots are minor but worth noting. The exact hypotheses on the domain groups in the general theorem are not fully detailed in the abstract, so one has to see the paper to judge how broad the result really is. The large cardinal equivalence is interesting within its niche, but measurable cardinals are a strong assumption, and the result is an equivalence rather than a proof of existence. No obvious circularity or fitting issues appear from the description.

This work is for specialists in infinite group theory, particularly those looking at topological properties and connections to set theory. A reader familiar with Polish groups and residually finite groups would get value from the automatic continuity part, while set theorists might focus on the ultraproduct claim.

I think it deserves serious peer review. The results are new and specific enough that referees can engage with the proof of the general theorem and check the applications.

Referee Report

0 major / 0 minor

Summary. The manuscript proves a general theorem constraining homomorphisms from certain topological groups to inverse limits of bounded-torsion groups. As corollaries, it establishes that every homomorphism from a Polish group to a countable torsion-free residually finite group has open kernel, and that the Grigorchuk group is a homomorphic image of a nonprincipal ultraproduct of groups if and only if a measurable cardinal exists.

Significance. If the central theorem is established, the results strengthen automatic-continuity statements for Polish groups and supply a set-theoretic characterization for a homomorphic-image property of the Grigorchuk group. These are standard topics in the area; the general theorem supplies a unified source for the two applications.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, assessment of significance, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper states a general theorem constraining homomorphisms from specified topological groups to inverse limits of bounded-torsion groups, from which the Polish-group automatic-continuity corollary and the Grigorchuk-ultraproduct equivalence follow directly. No equations or steps reduce by construction to inputs, no fitted parameters are renamed as predictions, and no load-bearing self-citations or uniqueness theorems imported from the authors' prior work appear in the abstract or described results. The derivation is self-contained against external benchmarks in topological group theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no information on free parameters, axioms, or invented entities can be extracted.

pith-pipeline@v0.9.1-grok · 5589 in / 1016 out tokens · 31596 ms · 2026-07-03T01:23:14.329375+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

27 extracted references

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O. Bogopolski and S. M. Corson,Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups, Mathematische Annalen 384 (2022), 1017–1055. HOMOMORPHISMS FROM TOPOLOGICAL GROUPS TO INVERSE LIMITS 11

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S. M. Corson,Automatic continuity ofℵ 1-free groups, Israel Journal of Mathematics 237 (2020), 267–285

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S. M. Corson,On homomorphic images of ultraproducts, J. Algebra 681 (2025), 306 – 317

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S. M. Corson, O. Varghese,A Nunke type classification in the locally compact setting, J. Algebra 563 (2020), 49 – 52

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Eda,On a Boolean power of a torsion free abelian group, J

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R. Grigorchuk, A. ˙Zuk,On a torsion-free weakly branch group defined by a three state au- tomaton, Internat. J. Algebra Comput. 12 (2002), 223 – 246

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M. Hruˇ s´ ak, J. van Mill, U. Ramos-Garc´ ıa, S. Shelah,Countably compact groups without non-trivial convergent sequences, Trans. Amer. Math. Soc. v374(2021), 1277-1296

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P. M¨ oller, L. Paris, and O. Varghese,On parabolic subgroups of Artin groups, Israel Journal of Mathematics 261 (2024), 809–840

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A. Prze´ zdziecki,Measurable cardinals and fundamental groups of compact spaces, Fund. Math. 192 (2006), 87 – 92. Brigham Young University, Department of Mathematics, Provo, UT 84602, USA Email address:conner@math.byu.edu E. T. S. I. I. Universidad Polit ´ecnica de Madrid, Jos ´e Guti ´errez Abascal 2, 28006 Madrid, Spain Email address:sammyc973@gmail.com...

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[1] [1]

Bartholdi, R

L. Bartholdi, R. I. Grigorchuk, Z. ˇSuni´k,Branch groups, Handbook of Algebra, Vol. 3, North- Holland 2003, 989 – 112

2003

[2] [2]

G. M. Bergman,Homomorphisms on infinite direct products of groups, rings and monoids, Pacific J. Math. 274 (2015), 451 – 495

2015

[3] [3]

Bogopolski and S

O. Bogopolski and S. M. Corson,Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups, Mathematische Annalen 384 (2022), 1017–1055. HOMOMORPHISMS FROM TOPOLOGICAL GROUPS TO INVERSE LIMITS 11

2022

[4] [4]

J. W. Cannon, G. R. Conner,The combinatorial structure of the Hawaiian earring group, Topology Appl. 106 (2000), 225 – 271

2000

[5] [5]

J. W. Cannon, G. R. Conner,On the fundamental groups of one-dimensional spaces, Topology Appl. 153 (2006), 2648 – 2672

2006

[6] [6]

G. R. Conner, S. M. Corson,A note on automatic continuity, Proc. Amer. Math. Soc. 147 (2019), 1255 –1268

2019

[7] [7]

S. M. Corson,Automatic continuity ofℵ 1-free groups, Israel Journal of Mathematics 237 (2020), 267–285

2020

[8] [8]

S. M. Corson,On homomorphic images of ultraproducts, J. Algebra 681 (2025), 306 – 317

2025

[9] [9]

S. M. Corson, O. Varghese,A Nunke type classification in the locally compact setting, J. Algebra 563 (2020), 49 – 52

2020

[10] [10]

Eda,On a Boolean power of a torsion free abelian group, J

K. Eda,On a Boolean power of a torsion free abelian group, J. Algebra 82 (1983), 84 – 93

1983

[11] [11]

Eda, Freeσ-products and non-commutatively slender groups, J

K. Eda, Freeσ-products and non-commutatively slender groups, J. Algebra148(1992), no. 1, 243–263

1992

[12] [12]

K. Eda, S. Shelah,The non-commutative Specker phenomenon in the uncountable case, J. Algebra 252 (2002), 22 – 26

2002

[13] [13]

Fuchs, Abelian Groups, Springer 2015

L. Fuchs, Abelian Groups, Springer 2015

2015

[14] [14]

G¨ obel, On stout and slender groups, J

R. G¨ obel, On stout and slender groups, J. Algebra35(1975), 39–55

1975

[15] [15]

Grigorchuk,On Burnside’s problem on periodic groups, Funktsional

R. Grigorchuk,On Burnside’s problem on periodic groups, Funktsional. Anal. i Prilozhen. 14 (1980), 53 –54, 1980

1980

[16] [16]

Grigorchuk, A

R. Grigorchuk, A. ˙Zuk,On a torsion-free weakly branch group defined by a three state au- tomaton, Internat. J. Algebra Comput. 12 (2002), 223 – 246

2002

[17] [17]

Hruˇ s´ ak, J

M. Hruˇ s´ ak, J. van Mill, U. Ramos-Garc´ ıa, S. Shelah,Countably compact groups without non-trivial convergent sequences, Trans. Amer. Math. Soc. v374(2021), 1277-1296

2021

[18] [18]

Jech, Set Theory: The Third Millenium Edition, Revised and Expanded, Springer 2000

T. Jech, Set Theory: The Third Millenium Edition, Revised and Expanded, Springer 2000

2000

[19] [19]

J. E. Keesling, Yu. B. Rudyak,On fundamental groups of compact Hausdorff spaces, Proc. Amer. Math. Soc. 135 (2007), 2629 – 2631

2007

[20] [20]

Keppeler, P

D. Keppeler, P. M¨ oller, O. Varghese,Automatic continuity for groups whose torsion subgroups are small, J. Group Theory 25 (2022), 1017 – 1043

2022

[21] [21]

Kramer and O

L. Kramer and O. Varghese,Abstract homomorphisms from locally compact groups to discrete groups, J. Algebra 538 (2019), 127–139

2019

[22] [22]

M¨ oller, L

P. M¨ oller, L. Paris, and O. Varghese,On parabolic subgroups of Artin groups, Israel Journal of Mathematics 261 (2024), 809–840

2024

[23] [23]

M¨ oller and O

P. M¨ oller and O. Varghese,Abstract group actions of locally compact groups on CAT(0) spaces, Groups, Geometry, and Dynamics 16 (2022), 863–887

2022

[24] [24]

Y. N. Moschovakis, Descriptive Set Theory: Second Edition, American Mathematical Society 2009

2009

[25] [25]

Nikolov, D

N. Nikolov, D. Segal,Generators and commutators in finite groups; abstract quotients of compact groups, Invent. Math. 190 (2012), 513 – 602

2012

[26] [26]

R. J. Nunke, Slender groups, Bull. Amer. Math. Soc.67(1961), 274–275

1961

[27] [27]

Prze´ zdziecki,Measurable cardinals and fundamental groups of compact spaces, Fund

A. Prze´ zdziecki,Measurable cardinals and fundamental groups of compact spaces, Fund. Math. 192 (2006), 87 – 92. Brigham Young University, Department of Mathematics, Provo, UT 84602, USA Email address:conner@math.byu.edu E. T. S. I. I. Universidad Polit ´ecnica de Madrid, Jos ´e Guti ´errez Abascal 2, 28006 Madrid, Spain Email address:sammyc973@gmail.com...

2006