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arxiv: 2605.23806 · v1 · pith:UBFUDMSDnew · submitted 2026-05-22 · 🧮 math.GR · math.LO

The geometrisation problem for topological groups

Christian Rosendal This is my paper

Pith reviewed 2026-05-25 02:32 UTC · model grok-4.3

classification 🧮 math.GR math.LO
keywords topological groupsPolish groupscoarse structureuniform structureLipschitz structurequasimetricgeometrisationhomeomorphism groups
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The pith

Topological groups carry intrinsic local Lipschitz and quasimetric structures from their left uniform and coarse data alone.

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

The paper develops a framework that assigns geometric structures to topological groups using only their given topology and group operation. It splits the geometry into a small-scale local Lipschitz category extracted from the left uniform structure and a large-scale quasimetric category extracted from the left coarse structure. For Polish groups the left coarse structure is metrisable precisely when the group is locally bounded, admits countable coverings by bounded sets, and possesses a compatible coarsely proper left-invariant metric. Minimal metrics then determine the local Lipschitz structure while maximal metrics determine the quasimetric structure; when both exist they combine into one canonical Lipschitz structure. The same framework is applied to homeomorphism groups, non-Archimedean Polish groups, and automorphism groups of Fraïssé limits.

Core claim

The geometrisation of a topological group splits into local Lipschitz and quasimetric categories definable from the canonical left uniform and left coarse structures respectively; for Polish groups, metrisability of the left coarse structure holds precisely when the group is locally bounded, admits countable coverings by bounded sets, and has a compatible coarsely proper left-invariant metric, with minimal metrics determining the local Lipschitz structure and maximal metrics determining the quasimetric structure.

What carries the argument

The left uniform structure and left coarse structure of the topological group, from which the local Lipschitz and quasimetric categories are defined, together with the minimal and maximal metrics that characterise each.

If this is right

  • When both structures exist they combine into a single canonical Lipschitz structure.
  • Minimal metrics determine the local Lipschitz structure.
  • Maximal metrics determine the quasimetric structure.
  • The framework applies to homeomorphism groups, non-Archimedean Polish groups, and automorphism groups of Fraïssé limits.

Where Pith is reading between the lines

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

  • The scale separation allows local and global geometric features of Polish groups to be studied independently without an external metric.
  • Similar intrinsic geometrisation might be attempted for topological groups outside the Polish setting.
  • The characterisations of minimal and maximal metrics make concrete computation of the structures possible in explicit examples.

Load-bearing premise

The canonical left uniform structure and left coarse structure already encode all data needed to define the local Lipschitz and quasimetric categories intrinsically.

What would settle it

A Polish group that is locally bounded and admits countable coverings by bounded sets, yet whose left coarse structure is not metrisable by any left-invariant metric.

read the original abstract

This paper presents a framework for assigning intrinsic geometric structures to topological groups using only the data provided by their topological and algebraic structure. The geometrisation spits into small-scale and large-scale components, formalised respectively through local Lipschitz and quasimetric categories that, in turn, are definable from the canonical left uniform and left coarse structures of the group. For Polish groups, the paper characterises metrisability of the left coarse structure in terms of local boundedness, countable coverings by bounded sets, and the existence of compatible coarsely proper left-invariant metrics. It then introduces minimal metrics, which determine local Lipschitz structure, and maximal metrics, which determine quasimetric structure, and provides intrinsic characterisations of both. When both structures exist, they combine into a single canonical Lipschitz structure. Our framework is subsequently applied to specific examples such as homeomorphism groups, non-Archimedean Polish groups and automorphism groups of Fra\"iss\'e limits.

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 / 1 minor

Summary. This paper presents a framework for assigning intrinsic geometric structures to topological groups using only their topological and algebraic data. The geometrisation splits into small-scale (local Lipschitz categories) and large-scale (quasimetric categories) components, both definable from the canonical left uniform and left coarse structures. For Polish groups, it characterizes the metrisability of the left coarse structure in terms of local boundedness, countable coverings by bounded sets, and the existence of compatible coarsely proper left-invariant metrics. It introduces minimal metrics determining the local Lipschitz structure and maximal metrics determining the quasimetric structure, with intrinsic characterisations, and combines them into a canonical Lipschitz structure when both exist. The framework is applied to homeomorphism groups, non-Archimedean Polish groups, and automorphism groups of Fraïssé limits.

Significance. If the claims hold with complete proofs, this provides a canonical intrinsic geometrisation of topological groups that could unify aspects of geometric group theory and topological dynamics. The metrisability characterization for Polish groups and the minimal/maximal metric constructions (derived directly from uniform/coarse data) are potentially valuable contributions, as are the applications to concrete classes such as homeomorphism groups.

minor comments (1)
  1. [Abstract] Abstract: 'The geometrisation spits into' is a typographical error and should be 'splits into'.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript and the recommendation of minor revision. No specific major comments appear in the report, so there are no individual points requiring a point-by-point response.

Circularity Check

0 steps flagged

No circularity: derivation self-contained from canonical uniform/coarse data

full rationale

The paper defines local Lipschitz and quasimetric categories directly from the standard left uniform and left coarse structures of a topological group (abstract and introduction). The metrisability characterisation for Polish groups is stated as a theorem in terms of local boundedness and existence of coarsely proper metrics, not as a definitional equivalence. Minimal/maximal metrics are constructed and characterised intrinsically without reducing to fitted parameters or self-citations. No load-bearing step quotes a prior result by the same author as an unverified uniqueness theorem, nor renames an input as a prediction. The framework is therefore independent of its target outputs and scores 0.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only access yields no explicit free parameters, axioms, or invented entities; full manuscript required to audit the ledger.

pith-pipeline@v0.9.0 · 5678 in / 1184 out tokens · 20278 ms · 2026-05-25T02:32:58.254127+00:00 · methodology

discussion (0)

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

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Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

  1. [1]

    Hiroshi Ando, Michal Doucha, and Yasumichi Matsuzawa,Large scale geometry of Banach- Lie groups, Trans. Amer. Math. Soc.375(2022), no. 4, 2827–2881, DOI 10.1090/tran/8576. MR4391735

    work page doi:10.1090/tran/8576 2022

  2. [2]

    Juliette Bavard, Spencer Dowdall, and Kasra Rafi,Isomorphisms between big mapping class groups, Int. Math. Res. Not. IMRN10(2020), 3084–3099, DOI 10.1093/imrn/rny093. MR4098634

    work page doi:10.1093/imrn/rny093 2020

  3. [3]

    MR1556955

    Garrett Birkhoff,A note on topological groups, Compositio Math.3(1936), 427–430. MR1556955

    work page 1936

  4. [4]

    Edwards and Robion C

    Robert D. Edwards and Robion C. Kirby,Deformations of spaces of imbeddings, Ann. of Math. (2)93(1971), 63–88, DOI 10.2307/1970753. MR0283802

    work page doi:10.2307/1970753 1971

  5. [5]

    Fisher,On the group of all homeomorphisms of a manifold, Trans

    Gordon M. Fisher,On the group of all homeomorphisms of a manifold, Trans. Amer. Math. Soc.97(1960), 193–212, DOI 10.2307/1993298. MR0117712

    work page doi:10.2307/1993298 1960

  6. [6]

    Math.48(2018), no

    Jes´ us Hern´andez Hern´andez, Israel Morales, and Ferr´an Valdez,Isomorphisms between curve graphs of infinite-type surfaces are geometric, Rocky Mountain J. Math.48(2018), no. 6, 1887–1904, DOI 10.1216/rmj-2018-48-6-1887. MR3879307

    work page doi:10.1216/rmj-2018-48-6-1887 2018

  7. [7]

    MR1462612

    Wilfrid Hodges,A shorter model theory, Cambridge University Press, Cambridge, 1997. MR1462612

    work page 1997

  8. [8]

    Shizuo Kakutani, ¨Uber die Metrisation der topologischen Gruppen, Proc. Imp. Acad. Tokyo 12(1936), no. 4, 82–84 (German). MR1568424

    work page 1936

  9. [9]

    Topol.27(2023), no

    Kathryn Mann and Kasra Rafi,Large-scale geometry of big mapping class groups, Geom. Topol.27(2023), no. 6, 2237–2296, DOI 10.2140/gt.2023.27.2237. MR4634747

    work page doi:10.2140/gt.2023.27.2237 2023

  10. [10]

    Large-scale geometry of homeomorphism groups

    Kathryn Mann and Christian Rosendal,Large-scale geometry of homeomorphism groups, Ergodic Theory Dynam. Systems38(2018), no. 7, 2748–2779, DOI 10.1017/etds.2017.8. MR3846725

    work page doi:10.1017/etds.2017.8 2018

  11. [11]

    Topol.18 (2014), no

    Emmanuel Militon,Distortion elements for surface homeomorphisms, Geom. Topol.18 (2014), no. 1, 521–614, DOI 10.2140/gt.2014.18.521. MR3159168

    work page doi:10.2140/gt.2014.18.521 2014

  12. [12]

    31, American Math- ematical Society, Providence, RI, 2003

    John Roe,Lectures on coarse geometry, University Lecture Series, vol. 31, American Math- ematical Society, Providence, RI, 2003. MR2007488

    work page 2003

  13. [13]

    2, 299–332, DOI 10.1515/FORUM.2009.014

    Christian Rosendal,A topological version of the Bergman property, Forum Math.21(2009), no. 2, 299–332, DOI 10.1515/FORUM.2009.014. MR2503307

    work page doi:10.1515/forum.2009.014 2009

  14. [14]

    Mat.56(2018), no

    ,Lipschitz structure and minimal metrics on topological groups, Ark. Mat.56(2018), no. 1, 185–206, DOI 10.4310/ARKIV.2018.v56.n1.a11. MR3800465

    work page doi:10.4310/arkiv.2018.v56.n1.a11 2018

  15. [15]

    223, Cambridge University Press, Cambridge, 2022

    ,Coarse geometry of topological groups, Cambridge Tracts in Mathematics, vol. 223, Cambridge University Press, Cambridge, 2022. MR4327092

    work page 2022

  16. [16]

    ,Geometries of topological groups, Bull. Amer. Math. Soc. (N.S.)60(2023), no. 4, 539–568, DOI 10.1090/bull/1807. MR4642118

    work page doi:10.1090/bull/1807 2023

  17. [17]

    Dyn.18(2024), no

    Anschel Schaffer-Cohen,Graphs of curves and arcs quasi-isometric to big mapping class groups, Groups Geom. Dyn.18(2024), no. 2, 705–735, DOI 10.4171/ggd/751. MR4729823 35 36 Christian Rosendal

    work page doi:10.4171/ggd/751 2024

  18. [18]

    153, American Mathematical Society, Providence, RI, 2014

    Terence Tao,Hilbert’s fifth problem and related topics, Graduate Studies in Mathematics, vol. 153, American Mathematical Society, Providence, RI, 2014. MR3237440

    work page 2014

  19. [19]

    Katrin Tent and Martin Ziegler,On the isometry group of the Urysohn space, J. Lond. Math. Soc. (2)87(2013), no. 1, 289–303, DOI 10.1112/jlms/jds027. MR3022717

    work page doi:10.1112/jlms/jds027 2013

  20. [20]

    Andr ´e Weil,L’int´egration dans les groupes topologiques et ses applications, Actualit ´es Sci- entifiques et Industrielles [Current Scientific and Industrial Topics], vol. No. 869, Hermann & Cie, Paris, 1940 (French). [This book has been republished by the author at Princeton, N. J., 1941.] MR0005741

    work page 1940