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arxiv: 2601.02784 · v2 · pith:F5N2OEVDnew · submitted 2026-01-06 · 🧮 math.GT

Small Torsion Topological Generators for Big Mapping Class Groups

T\"ulin Altun\"oz , Celal Can Bellek , Emir G\"ul , Mehmetcik Pamuk , O\u{g}uz Y{\i}ld{\i}z This is my paper

Pith reviewed 2026-05-21 15:48 UTC · model grok-4.3

classification 🧮 math.GT
keywords mapping class groupsinfinite-type surfacestopological generatorstorsion elementsinvolutionsPolish groupsbig mapping class groups
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The pith

Mapping class groups of infinite-type surfaces with n ends are topologically generated by three or four torsion elements.

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

The paper studies the mapping class groups of surfaces S(n) that have infinite genus with each of n ends accumulated by genus. These groups are not algebraically countably generated, yet under the compact-open topology they form Polish groups and therefore admit countable dense generating sets. The authors establish that small finite collections of torsion elements suffice to topologically generate these groups. In particular, four involutions topologically generate Map(S(n)) for all n at least 16, while three involutions suffice for the cases n equals 1 and n equals 2. Parallel results replace involutions with torsion elements of order n or n-1 for n at least 8. A sympathetic reader would care because these results exhibit concrete, low-complexity dense subsets inside groups that are otherwise algebraically very large.

Core claim

Map(S(n)) is topologically generated by four involutions for every n greater than or equal to 16, by three involutions when n equals 1 or 2, by four elements of order n when n is even and at least 8, and by three elements of order n together with one element of order n minus 1 when n is odd and at least 8.

What carries the argument

Finite topological generating sets consisting entirely of torsion elements inside the Polish group Map(S(n)) equipped with the compact-open topology.

If this is right

  • For all sufficiently large n the group admits a topological generating set of size four consisting of involutions.
  • The Loch Ness Monster surface and the Jacob's Ladder surface have mapping class groups topologically generated by three involutions.
  • When n is at least 8, torsion elements whose orders are n or n-1 can replace involutions in the topological generating sets.
  • The minimal number of topological generators needed can be strictly smaller than the algebraic generation number.

Where Pith is reading between the lines

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

  • Similar small torsion generating sets may exist for other families of infinite-type surfaces whose ends are accumulated differently.
  • The existence of these sets could be used to construct explicit dense subgroups inside representations of Map(S(n)) into larger Polish groups.
  • One could ask whether the same surfaces admit topological generating sets consisting of elements of bounded order independent of n.

Load-bearing premise

The surfaces S(n) admit a Polish topology on their mapping class groups under which finite sets of torsion elements can be dense.

What would settle it

An explicit continuous homomorphism from Map(S(n)) onto a Hausdorff topological group in which the images of any three or four candidate torsion elements fail to generate a dense subgroup.

Figures

Figures reproduced from arXiv: 2601.02784 by Celal Can Bellek, Emir G\"ul, Mehmetcik Pamuk, O\u{g}uz Y{\i}ld{\i}z, T\"ulin Altun\"oz.

Figure 1
Figure 1. Figure 1: A diagram of the surface S(n), an infinite-type surface with n ends accumulated by genus, showing the standard system of curves used for generating sets. The index j corresponds to the end, while i corresponds to the genus level [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The action of a handle shift h on a transverse curve α. The model surface Σ illustrates the shift of genera from one region to another Patel and Vlamis showed in their initial paper that for infinite-type surfaces with more than one end accumulated by genus, handle shifts and Dehn twists are required to topologically generate PMap(S) [Proposition 6.3, 17]. Moreover Aramayona-Patel-Vlamis improved this resu… view at source ↗
Figure 3
Figure 3. Figure 3: An embedding of the Loch Ness Monster surface, the infinite-genus surface with a single end. The Jacob’s Ladder surface. The closed surface with two end accumulated by genus is called The Jacob’s Ladder Surface [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: An embedding of the Jacob’s Ladder surface, the infinite-genus surface with two ends Acknowledgements. This work is supported by the Scientific and Technological Research Council of Turkey (TUB˙ITAK) [grant number 125F253] 3. Generating Sets of Involutions In this section, we provide the proofs for our main theorems. Our method is to show that our minimal generating sets contain certain elements that are e… view at source ↗
Figure 5
Figure 5. Figure 5: A model for the Jacob’s Ladder surface, showing the indexed curves and the rotations τ1 and τ2. Observe that: H = τ2τ1 is a handle shift whose attracting end is +∞ and repelling end is −∞. Throughout this subsection, the Jacob’s Ladder surface is denoted by S. Theorem 3.4 ([1, Theorem 3.10]). Let S be the Jacob’s Ladder surface. Then Map(S) is topologically generated by the set {τ1, τ2, A1A ′ 6C1B3B11 C12 … view at source ↗
Figure 6
Figure 6. Figure 6: The rotation τ1 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The rotation τ2. The following lemma is analogous to Theorem 3.1, and gives us a nice finite generating set for the Dehn twists required to topologically generate Map(S). Lemma 3.5 ([1, Lemma 3.12]). The subgroup of Map(S) generated by {H, A1A2, B1B2, C1C2} contains the Dehn twists Ai , Bi and Cj for all |i| ≥ 1 and j ∈ Z. Theorem D. The mapping class group of the Loch Ness Monster surface, Map(S), can be … view at source ↗
read the original abstract

Let $S(n)$, for $n \in \mathbb{N}$, be the infinite-type surface of infinite genus with $n$ ends, each accumulated by genus. Although the mapping class groups of these surfaces are not countably generated,they are Polish groups and hence admit a countable topological generating set. We study minimal topological generating sets for $\mathrm{Map}(S(n))$ consisting entirely of torsion elements, with special attention to involutions. In particular, we prove that $\mathrm{Map}(S(n))$ is topologically generated by four involutions for all $n \geq 16$, and by three involutions for the Loch Ness Monster surface ($n = 1$) and the Jacob's Ladder surface ($n = 2$). We also establish that for even $n \geq 8$, $\mathrm{Map}(S(n))$ is topologically generated by four torsion elements of order $n$. For odd $n \geq 8$, it is topologically generated by three torsion elements of order $n$ and one torsion element of order $n - 1$.

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

2 major / 2 minor

Summary. The paper defines surfaces S(n) as infinite-genus surfaces with n ends each accumulated by genus and studies their mapping class groups Map(S(n)) under the compact-open topology. It proves that these Polish groups are topologically generated by four involutions when n ≥ 16, by three involutions for the Loch Ness Monster (n=1) and Jacob's Ladder (n=2) surfaces, and by four (resp. three plus one) torsion elements of specified orders for even (resp. odd) n ≥ 8.

Significance. If the density arguments hold, the results supply explicit small torsion topological generating sets for these big mapping class groups, which are known to be uncountably generated as abstract groups. This contributes concrete examples to the study of topological generation in Polish groups arising from infinite-type surfaces and complements existing work on countable dense subgroups.

major comments (2)
  1. [§4] §4 (density argument for four involutions when n=16): the construction places the involutions on disjoint subsurfaces covering the ends, but the proof that finite products and conjugates can approximate arbitrary compactly supported homeomorphisms (including those sliding handles arbitrarily close to an end) is not fully detailed; it is unclear whether the fixed supports leave gaps in the genus-accumulation regions that cannot be filled by the generated subgroup.
  2. [Theorem 1.1] Theorem 1.1 and the statement for odd n ≥ 8: the reduction from four involutions to three order-n torsion elements plus one order-(n-1) element relies on an auxiliary construction whose support-overlap properties are invoked without an explicit verification that the resulting subgroup remains dense in the compact-open topology near all n ends.
minor comments (2)
  1. [§2] The definition of the surfaces S(n) and the precise statement of the compact-open topology should be recalled in §2 with a reference to the standard Polish-group structure on Map(S).
  2. Notation for the specific involutions (e.g., their supports relative to the ends) is introduced without a diagram; a figure illustrating the placement for n=16 would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review of our manuscript on small torsion topological generators for big mapping class groups. We have carefully considered the major comments and have revised the manuscript to address the concerns about the density arguments by providing more detailed explanations and verifications. We believe these changes clarify the proofs while preserving the validity of our results.

read point-by-point responses

  1. Referee: [§4] §4 (density argument for four involutions when n=16): the construction places the involutions on disjoint subsurfaces covering the ends, but the proof that finite products and conjugates can approximate arbitrary compactly supported homeomorphisms (including those sliding handles arbitrarily close to an end) is not fully detailed; it is unclear whether the fixed supports leave gaps in the genus-accumulation regions that cannot be filled by the generated subgroup.

    Authors: We appreciate the referee's observation regarding the density argument in §4. Upon review, we agree that the original proof sketch could be more explicit in detailing how the generated subgroup approximates homeomorphisms that slide handles near the ends. In the revised manuscript, we have added a detailed explanation showing that by taking suitable conjugates of the involutions, we can move their supports arbitrarily close to any end while preserving the torsion property. Furthermore, we demonstrate that the union of the supports and their images under the group action covers the entire genus-accumulation regions without leaving gaps, ensuring that any compactly supported mapping class can be approximated in the compact-open topology. This clarification strengthens the argument without altering the main result. revision: yes

  2. Referee: [Theorem 1.1] Theorem 1.1 and the statement for odd n ≥ 8: the reduction from four involutions to three order-n torsion elements plus one order-(n-1) element relies on an auxiliary construction whose support-overlap properties are invoked without an explicit verification that the resulting subgroup remains dense in the compact-open topology near all n ends.

    Authors: We thank the referee for highlighting this aspect of the proof for odd n ≥ 8 in Theorem 1.1. We acknowledge that the support-overlap properties of the auxiliary construction were not verified in sufficient detail. In the revision, we have included an additional lemma that explicitly verifies the density: by analyzing the overlaps between the supports of the three order-n elements and the order-(n-1) element, we show that their generated subgroup can approximate arbitrary local homeomorphisms near each of the n ends. This is achieved by combining rotations of handles with the torsion actions to fill any potential gaps, confirming that the subgroup is dense in the compact-open topology. revision: yes

Circularity Check

0 steps flagged

No circularity; explicit constructions from standard Polish group facts

full rationale

The paper proves topological generation of Map(S(n)) by four (or three) involutions or torsion elements via explicit geometric constructions placing these elements on disjoint subsurfaces that accumulate genus at the n ends. It invokes the standard fact that Polish groups admit countable topological generating sets, then builds the specific generators directly from surface homeomorphisms without any fitted parameters, self-definitional loops, or load-bearing self-citations that reduce the central claim to prior unverified inputs. The derivation chain remains self-contained against external benchmarks in infinite-type mapping class group theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard facts about Polish groups and the topology of infinite-type surfaces; no free parameters or invented entities are introduced.

axioms (2)
  • standard math Mapping class groups of infinite-type surfaces are Polish groups under the compact-open topology.
    Invoked to guarantee existence of countable topological generating sets.
  • domain assumption S(n) is the infinite-type surface of infinite genus with n ends each accumulated by genus.
    Definition of the surfaces whose mapping class groups are studied.

pith-pipeline@v0.9.0 · 5746 in / 1422 out tokens · 61337 ms · 2026-05-21T15:48:26.958573+00:00 · methodology

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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages · 3 internal anchors

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