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The liftable mapping class group of balanced superelliptic covers

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Tyrone Ghaswala, Rebecca R. Winarski

Geometric Topology · Submitted 13 Apr 2016

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We compute the index of the liftable mapping class group in the full mapping class group of the sphere and show that the liftable mapping class group is independent of the degree of the cover.

how it is stated · asserted 1.00 read the record →

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arXiv ID1604.03908 (version 4)
Pith IDSYJRIHW2, Pith's permanent record of this paper and its review
CitationRebecca R. Winarski, Tyrone Ghaswala. "The liftable mapping class group of balanced superelliptic covers." arXiv:1604.03908 (2016). https://pith.science/paper/SYJRIHW2
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,395 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.87 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.37 · compares with prior results 0.63 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; .

The hyperelliptic mapping class group has been studied in various contexts within topology and algebraic geometry. What makes this study tractable is that there is a surjective map from the hyperelliptic mapping class group to a mapping class group of a punctured sphere. The more general family of superelliptic mapping class groups does not, in general, surject on to a mapping class group of a punctured sphere, but on to a finite index subgroup. We call this finite index subgroup the liftable mapping class group. In order to initiate the generalization of results on the hyperelliptic mapping class group to the broader family of superelliptic mapping class groups, we study an intermediate family called the balanced superelliptic mapping class group. We compute the index of the liftable mapping class group in the full mapping class group of the sphere and show that the liftable mapping class group is independent of the degree of the cover. We also build a presentation for the liftable mapping class group, compute its abelianization, and show that the balanced superelliptic mapping class group has finite abelianization. Although our calculations focus on the subfamily of balanced superelliptic mapping class groups, our techniques can be extended to any superelliptic mapping class group, even those not within the balanced family.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 The hyperelliptic mapping class group has been studied in various contexts within topology and algebraic geometry. background 1.00 0.03 0.02 asserted 1.00
s1 What makes this study tractable is that there is a surjective map from the hyperelliptic mapping class group to a mapping class group of a punctured sphere. background 0.85 0.09 0.04 asserted 1.00
s2 The more general family of superelliptic mapping class groups does not, in general, surject on to a mapping class group of a punctured sphere, but on to a finite index subgroup. background 0.86 0.22 0.08 asserted 0.99
s3 We call this finite index subgroup the liftable mapping class group. background 0.61 0.55 0.05 asserted 0.99
s4 In order to initiate the generalization of results on the hyperelliptic mapping class group to the broader family of superelliptic mapping class groups, we study an intermediate family called the balanced superelliptic mapping class group. contribution 0.84 0.66 0.05 asserted 0.98
s5 We compute the index of the liftable mapping class group in the full mapping class group of the sphere and show that the liftable mapping class group is independent of the degree of the cover. result 0.99 0.96 0.31 asserted 1.00
s6 We also build a presentation for the liftable mapping class group, compute its abelianization, and show that the balanced superelliptic mapping class group has finite abelianization. result 0.80 0.96 0.28 asserted 1.00
s7 Although our calculations focus on the subfamily of balanced superelliptic mapping class groups, our techniques can be extended to any superelliptic mapping class group, even those not within the balanced family. interpretation 0.43 0.53 0.12 asserted 0.51

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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not examined
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  • references and citing sentences
  • relations to other papers
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  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

The hyperelliptic mapping class group has been studied in various contexts within topology and algebraic geometry. What makes this study tractable is that there is a surjective map from the hyperelliptic mapping class group to a mapping class group of a punctured sphere. The more general family of superelliptic mapping class groups does not, in general, surject on to a mapping class group of a punctured sphere, but on to a finite index subgroup. We call this finite index subgroup the liftable mapping class group. In order to initiate the generalization of results on the hyperelliptic mapping class group to the broader family of superelliptic mapping class groups, we study an intermediate family called the balanced superelliptic mapping class group. We compute the index of the liftable mapping class group in the full mapping class group of the sphere and show that the liftable mapping class group is independent of the degree of the cover. We also build a presentation for the liftable mapping class group, compute its abelianization, and show that the balanced superelliptic mapping class group has finite abelianization. Although our calculations focus on the subfamily of balanced superelliptic mapping class groups, our techniques can be extended to any superelliptic mapping class group, even those not within the balanced family.

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3 arXiv papers cite this in 5 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Cites it as background 1 paper
  • In [ref] Ghaswala-Winarski studied a class of cyclic covers which they call balanced superelliptic covers; these are the cyclic covers that are branched over $2n+2$ points, where $n = g/(d-1)$.

    Picard groups of moduli spaces of curves with symmetry · Balanced Superelliptic Covers · 0.93

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