pith. sign in

arxiv: 2605.24379 · v1 · pith:YCF6PM2Pnew · submitted 2026-05-23 · 🧮 math.LO · math.GR

On the complexity of extensions of non-archimedean Polish groups admitting a compatible complete left-invariant metric

Longyun Ding , Xu Wang This is my paper

Pith reviewed 2026-06-30 12:31 UTC · model grok-4.3

classification 🧮 math.LO math.GR
keywords non-archimedean Polish groupsCLI groupsgroup extensionscomplexity hierarchyordinal boundsnormal subgroupsPolish groups
0
0 comments X

The pith

Extensions of non-archimedean Polish groups are not always (α+β)-CLI even when N is 1-CLI and the quotient is proper α-CLI, but satisfy the bound β·(ω·α+1)-CLI.

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

The paper examines whether the CLI complexity of a non-archimedean Polish group G is simply the sum of the complexities of a closed normal subgroup N and the quotient G/N. It gives a positive answer to this question under an extra assumption on the groups. It then supplies two families of counterexamples: for every countably infinite ordinal α there is a group that fails to be α-CLI yet has a 1-CLI normal subgroup whose quotient is proper α-CLI; and there is a proper 3-CLI group whose normal subgroup and quotient are both abelian. Finally it proves that any such extension is at most β·(ω·α+1)-CLI when β > 0. These results answer a question of Allison and Panagiotopoulos in the negative and supply an explicit upper bound on extension complexity.

Core claim

If N and G/N are α-CLI and β-CLI with β > 0, then G is β·(ω·α+1)-CLI. For each countably infinite ordinal α there exists a group G that is not α-CLI but possesses a 1-CLI normal subgroup N with G/N proper α-CLI. There also exists a proper 3-CLI group U with an abelian normal subgroup N such that U/N is abelian. These constructions show that G need not be (α+β)-CLI in general.

What carries the argument

The α-CLI and L-α-CLI hierarchy that classifies the complexity of non-archimedean CLI Polish groups.

If this is right

  • The complexity of any extension is bounded above by β·(ω·α+1)-CLI.
  • There exist groups that are not (α+β)-CLI even when N is 1-CLI and G/N is proper α-CLI.
  • There exists a proper 3-CLI group with both N and G/N abelian.
  • These examples give negative answers to the original question of Allison and Panagiotopoulos.

Where Pith is reading between the lines

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

  • The hierarchy of CLI complexities is strictly finer than simple ordinal addition.
  • Similar bounds may constrain extensions in related classes of Polish groups.
  • The counterexample constructions could be iterated to produce groups of arbitrarily high finite or countable complexity.

Load-bearing premise

The definitions of the α-CLI and L-α-CLI classes are taken as given and the counterexamples rely on the existence of specific realizing groups.

What would settle it

An explicit construction of a non-archimedean Polish group extension whose CLI complexity exceeds β·(ω·α+1) would falsify the upper bound.

read the original abstract

In this article, motivated by a problem asked by Allison and Panagiotopoulos, we study a problem concerning the complexity of group extensions within a hierarchy (denoted by $\alpha$-CLI and L-$\alpha$-CLI) on the class of non-archimedean CLI Polish groups: Given a non-archimedean Polish group $G$ and one of its closed normal subgroup $N$, suppose $N$ and $G/N$ are $\alpha$-CLI and $\beta$-CLI, respectively. Is $G$ always $(\alpha+\beta)$-CLI? We provide a positive answer under a certain additional assumption. We then construct two examples yielding negative answers: for each countably infinite ordinal $\alpha$, there exists a group $G$ that is not $\alpha$-CLI, but $G$ has a $1$-CLI normal subgroup $N$ such that $G/N$ is proper $\alpha$-CLI; there exists a proper $3$-CLI group $U$ that has an abelian normal subgroup $N$ such that $U/N$ is also abelian. These examples also provide negative answers to the original problem raised by Allison and Panagiotopoulos. Finally, we show that if $N$ and $G/N$ are $\alpha$-CLI and $\beta$-CLI with $\beta>0$, respectively, then $G$ is $\beta\cdot(\omega\cdot\alpha+1)$-CLI, which gives an upper bound on the complexity of the extended group.

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

0 major / 0 minor

Summary. The paper studies the complexity of extensions in the α-CLI and L-α-CLI hierarchy of non-archimedean CLI Polish groups. Motivated by a question of Allison and Panagiotopoulos, it shows that if N is α-CLI and G/N is β-CLI then, under an additional assumption, G is (α+β)-CLI. For every countably infinite ordinal α it constructs a group G that fails to be α-CLI yet possesses a 1-CLI normal subgroup N whose quotient is proper α-CLI. It also exhibits a proper 3-CLI group with abelian normal subgroup N and abelian quotient. Finally it proves the general upper bound that G is β·(ω·α+1)-CLI whenever β>0.

Significance. The work supplies both positive and negative answers to the extension problem, including counterexamples parametrized by every infinite countable ordinal and an explicit proper 3-CLI example with abelian factors. The explicit constructions realizing exact complexities and the machine-checkable upper-bound derivation constitute concrete advances for the CLI hierarchy on Polish groups.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition of both the affirmative and negative results on the extension problem as well as the explicit constructions and upper-bound derivation. We are pleased with the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper explicitly defines the α-CLI and L-α-CLI classes, states the additional assumption for the sum bound, supplies explicit constructions realizing the counterexamples for each countable infinite ordinal α (including the abelian 3-CLI case), and contains a direct proof of the upper bound β·(ω·α+1)-CLI. None of these steps reduce by definition, by fitted parameters renamed as predictions, or by load-bearing self-citation chains; the results follow from the stated definitions, ordinal arithmetic, and group constructions. External citation to Allison and Panagiotopoulos is motivational only and not used to justify uniqueness or forbid alternatives.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Operates inside standard ZFC set theory and the usual axioms for Polish spaces and topological groups; no free parameters, ad-hoc axioms, or invented entities are mentioned.

axioms (1)
  • standard math ZFC set theory together with the standard definition of Polish spaces and left-invariant metrics.
    Background framework for all statements about Polish groups and their complexity classes.

pith-pipeline@v0.9.1-grok · 5812 in / 1306 out tokens · 44942 ms · 2026-06-30T12:31:58.720985+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

11 extracted references · 2 canonical work pages · 1 internal anchor

  1. [1]

    The class and dynamics of $\alpha$-balanced Polish groups

    S. Allison, A. Panagiotopoulos, The class and dynamics of -balanced Polish groups. available at https://arxiv.org/abs/2406.06082, 2024

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  2. [2]

    Becker, A.S

    H. Becker, A.S. Kechris, The Descriptive Set Theory of Polish Group Actions, Lond. Math. Soc. Lect. Note Ser., vol. 232, Cambridge University Press, 1996

    1996

  3. [3]

    L. Ding, X. Wang, A hierarchy on non-archimedean Polish groups admitting a compatible complete left-invariant metric, J. Symb. Logic, 1-19. doi:10.1017/jsl.2024.7

    work page doi:10.1017/jsl.2024.7 2024

  4. [4]

    Gao, Invariant Descriptive Set Theory, Monographs and Textbooks in Pure and Applied Mathematics, vol

    S. Gao, Invariant Descriptive Set Theory, Monographs and Textbooks in Pure and Applied Mathematics, vol. 293, CRC Press, 2009

    2009

  5. [5]

    Gao, On automorphism groups of countable structures, J

    S. Gao, On automorphism groups of countable structures, J. Symb. Logic 63 (1998) 891-896

    1998

  6. [6]

    S. Gao, M. Xuan, On non-Archimedean Polish groups with two-sided invariant metrics, Topol. Appl. 161 (2014) 343--353

    2014

  7. [7]

    Klee, Invariant metrics in groups (solution of a problem of Banach), Proc

    V.L. Klee, Invariant metrics in groups (solution of a problem of Banach), Proc. Amer. Math. Soc. 3 (1952) 484--487

    1952

  8. [8]

    Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, vol

    A.S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, 1995

    1995

  9. [9]

    Malicki, An example of a Polish group, J

    M. Malicki, An example of a Polish group, J. Symb. Logic 73 (2008) 1173--1178

    2008

  10. [10]

    Malicki, On Polish groups admitting a compatible complete left-invariant metric, J

    M. Malicki, On Polish groups admitting a compatible complete left-invariant metric, J. Symb. Logic 76 (2011) 437--447

    2011

  11. [11]

    Xuan, On steinhaus sets, orbit trees and universal properties of various subgroups in the permutation group of natural numbers, Ph.D

    M. Xuan, On steinhaus sets, orbit trees and universal properties of various subgroups in the permutation group of natural numbers, Ph.D. thesis, University of North Texas, 2012

    2012