pith. sign in

arxiv: 2605.23822 · v1 · pith:ELPIP34Znew · submitted 2026-05-22 · 🧮 math.GR · math.GN· math.LO

Coarse Structures on Homogeneous Spaces

Carlos P\'erez Estrada , Christian Rosendal This is my paper

Pith reviewed 2026-05-25 02:30 UTC · model grok-4.3

classification 🧮 math.GR math.GNmath.LO
keywords coarse structureshomogeneous spacesPolish groupsmapping class groupsLoch Ness monster surfacequotient groupsleft coarse structurenormal subgroups
0
0 comments X

The pith

The left coarse structure on the quotient G/H does not always equal the quotient of the left coarse structure on G.

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 left coarse structure on a quotient group G/H coincides with the quotient of the left coarse structure on G when H is a closed normal subgroup of the topological group G. It constructs a counterexample in the class of Polish groups using the mapping class group of the Loch Ness monster surface as a quotient of the mapping class group of the punctured Loch Ness monster surface. The paper also derives equivalent and sufficient conditions for when the two structures coincide, formulated in terms of liftings of bounded sets, existence of transversals, and metrisability of the left coarse structure of G restricted to H.

Core claim

For a closed normal subgroup H of a topological group G, the left coarse structure on the quotient G/H does not necessarily equal the quotient of the left coarse structure on G. This is shown by a counterexample among Polish groups given by the mapping class group of the Loch Ness monster surface viewed as a quotient of the mapping class group of the punctured Loch Ness monster surface. Equivalent and sufficient conditions for equality are established in special settings using liftings of bounded sets, transversals, and metrisability.

What carries the argument

The direct comparison of the left coarse structure on the quotient group G/H against the quotient coarse structure induced from G, mediated by liftings of bounded sets and transversals.

If this is right

  • When bounded sets lift through the quotient map, the two coarse structures coincide.
  • Existence of a transversal for H in G implies equality of the structures under the left uniformity.
  • Metrisability of the left coarse structure of G restricted to H is sufficient for the structures to agree in certain cases.
  • The equality fails for the mapping class groups of the Loch Ness monster surfaces.

Where Pith is reading between the lines

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

  • Coarse invariants on homogeneous spaces of Polish groups may need separate treatment when the quotient map does not preserve boundedness in the expected way.
  • Similar discrepancies could appear in other infinite-dimensional Polish groups with complicated normal subgroups.
  • The conditions on liftings and transversals offer a practical checklist for verifying when quotient coarse structures can be computed directly from the ambient group.

Load-bearing premise

The mapping class groups of the Loch Ness monster surface and its punctured version are Polish groups whose left coarse structures produce unequal quotient structures.

What would settle it

An explicit construction of a transversal or a lifting of bounded sets showing that the two coarse structures coincide for these specific mapping class groups would falsify the counterexample.

read the original abstract

Given a closed normal subgroup $H$ of a topological group $G$, we address the question of whether the left coarse structure on the quotient group $G/H$ equals the quotient of the left coarse structure on $G$. We provide a counterexample among Polish groups, namely, the mapping class group of the Loch Ness monster surface seen as a quotient of the mapping class group of the punctured Loch Ness monster surface, and establish both equivalent and sufficient conditions for when this holds in special settings. The latter are formulated in terms of liftings of bounded sets, existence of transversals and metrisability of the left coarse structure of $G$ restricted to $H$.

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 / 0 minor

Summary. The paper investigates whether, for a closed normal subgroup H of a topological group G, the left coarse structure on the quotient G/H coincides with the quotient of the left coarse structure on G. It constructs a counterexample among Polish groups, taking the mapping class group of the Loch Ness monster surface as a quotient of the mapping class group of the punctured Loch Ness monster surface, and derives equivalent and sufficient conditions for equality in terms of liftings of bounded sets, existence of transversals, and metrisability of the restricted left coarse structure on H.

Significance. If the counterexample is correct, the result shows that the two natural coarse structures on homogeneous spaces can fail to coincide even when G and G/H are Polish groups, which is relevant to coarse geometry and the study of large-scale properties of topological groups. The listed conditions supply concrete criteria that may be applicable in other settings involving Polish groups or mapping class groups of infinite-type surfaces.

major comments (1)
  1. [Abstract (counterexample paragraph)] The central claim rests on the mapping class groups of the Loch Ness monster surface and its punctured version being Polish groups whose left coarse structures produce the stated inequality; without explicit verification of the Polish topology, the bounded sets, and the failure of the quotient coarse structure to match (as referenced in the abstract's counterexample paragraph), the load-bearing step cannot be confirmed from the supplied text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and for identifying the need for clearer verification of the central counterexample. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract (counterexample paragraph)] The central claim rests on the mapping class groups of the Loch Ness monster surface and its punctured version being Polish groups whose left coarse structures produce the stated inequality; without explicit verification of the Polish topology, the bounded sets, and the failure of the quotient coarse structure to match (as referenced in the abstract's counterexample paragraph), the load-bearing step cannot be confirmed from the supplied text.

    Authors: We agree that the manuscript as supplied does not contain a fully self-contained verification of the Polish topology, the identification of bounded sets, and the explicit demonstration that the left coarse structure on the quotient differs from the quotient coarse structure. While the argument draws on established facts about the Polish topology of mapping class groups of infinite-type surfaces, the referee is correct that these steps require explicit spelling out to be confirmable from the text alone. In the revised manuscript we will add a dedicated subsection (in the counterexample section) that recalls the relevant Polish topology, specifies the bounded sets via the left entourages, and constructs the sets witnessing the strict inequality. revision: yes

Circularity Check

0 steps flagged

No significant circularity; counterexample is an independent construction

full rationale

The paper's central claim is the existence of a counterexample (mapping class group of Loch Ness monster surface as quotient of the punctured version) showing that the left coarse structure on G/H need not equal the quotient coarse structure on G. This is presented as an explicit construction among Polish groups rather than a derivation from equations or parameters. Equivalent and sufficient conditions are stated separately in terms of liftings, transversals, and metrisability, without reducing to self-referential definitions or fitted inputs. No load-bearing self-citations or ansatzes are invoked in the abstract or described structure. The result is self-contained against external benchmarks via the concrete groups chosen.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review based on abstract only; no explicit free parameters, ad-hoc axioms, or invented entities are identifiable from the given text.

pith-pipeline@v0.9.0 · 5634 in / 1206 out tokens · 21579 ms · 2026-05-25T02:30:13.268868+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

17 extracted references · 17 canonical work pages

  1. [1]

    1981 , PAGES =

    Roelcke, Walter and Dierolf, Susanne , TITLE =. 1981 , PAGES =

    work page 1981

  2. [2]

    Kakutani, Shizuo and Kodaira, Kunihiko , title =. Proc. Imp. Acad. Tokyo , volume =. 1944 , pages =

    work page 1944

  3. [3]

    Militon, Emmanuel , TITLE =. Geom. Topol. , FJOURNAL =. 2014 , NUMBER =. doi:10.2140/gt.2014.18.521 , URL =

    work page doi:10.2140/gt.2014.18.521 2014

  4. [4]

    Large-scale geometry of homeomorphism groups

    Mann, Kathryn and Rosendal, Christian , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 2018 , NUMBER =. doi:10.1017/etds.2017.8 , URL =

    work page doi:10.1017/etds.2017.8 2018

  5. [5]

    , title =

    Struble, Raimond A. , title =. Compositio Mathematica , volume =. 1974 , pages =

    work page 1974

  6. [6]

    2022 , PAGES =

    Rosendal, Christian , TITLE =. 2022 , PAGES =

    work page 2022

  7. [7]

    Topology Appl

    Nicas, Andrew and Rosenthal, David , TITLE =. Topology Appl. , FJOURNAL =. 2012 , NUMBER =. doi:10.1016/j.topol.2012.06.009 , URL =

    work page doi:10.1016/j.topol.2012.06.009 2012

  8. [8]

    Groups Geom

    Zielinski, Joseph , TITLE =. Groups Geom. Dyn. , FJOURNAL =. 2021 , NUMBER =. doi:10.4171/ggd/628 , URL =

    work page doi:10.4171/ggd/628 2021

  9. [9]

    2003 , PAGES =

    Roe, John , TITLE =. 2003 , PAGES =. doi:10.1090/ulect/031 , URL =

    work page doi:10.1090/ulect/031 2003

  10. [10]

    Hill, Thomas , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 2025 , NUMBER =. doi:10.1090/proc/17181 , URL =

    work page doi:10.1090/proc/17181 2025

  11. [11]

    , TITLE =

    Kechris, Alexander S. , TITLE =. 1995 , PAGES =. doi:10.1007/978-1-4612-4190-4 , URL =

    work page doi:10.1007/978-1-4612-4190-4 1995

  12. [12]

    Rosendal, Christian , TITLE =. Bull. Symbolic Logic , FJOURNAL =. 2009 , NUMBER =. doi:10.2178/bsl/1243948486 , URL =

    work page doi:10.2178/bsl/1243948486 2009

  13. [13]

    Mann, Kathryn and Rafi, Kasra , TITLE =. Geom. Topol. , FJOURNAL =. 2023 , NUMBER =. doi:10.2140/gt.2023.27.2237 , URL =

    work page doi:10.2140/gt.2023.27.2237 2023

  14. [14]

    Domat, George and Hoganson, Hannah and Kwak, Sanghoon , TITLE =. Adv. Math. , FJOURNAL =. 2023 , PAGES =. doi:10.1016/j.aim.2022.108836 , URL =

    work page doi:10.1016/j.aim.2022.108836 2023

  15. [15]

    1977 , PAGES =

    Hogbe-Nlend, Henri , TITLE =. 1977 , PAGES =

    work page 1977

  16. [16]

    Topology Appl

    Dikranjan, Dikran and Zava, Nicol\`o , TITLE =. Topology Appl. , FJOURNAL =. 2017 , PAGES =. doi:10.1016/j.topol.2017.04.011 , URL =

    work page doi:10.1016/j.topol.2017.04.011 2017

  17. [17]

    On the lifting of bounded sets in

    Bonet, Jos. On the lifting of bounded sets in. Proceedings of the Edinburgh Mathematical Society , series =. 1993 , doi =

    work page 1993