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Alternating and symmetric actions on surfaces
Geometric Topology · Submitted 10 Oct 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In this article, we derive necessary and sufficient conditions under which two torsion elements in $\mathrm{Mod}(S_g)$ will have conjugates that generate a finite symmetric or an alternating subgroup of $\mathrm{Mod}(S_g)$.
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The abstract, sentence by sentence
Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface of genus $g \geq 2$. In this article, we derive necessary and sufficient conditions under which two torsion elements in $\mathrm{Mod}(S_g)$ will have conjugates that generate a finite symmetric or an alternating subgroup of $\mathrm{Mod}(S_g)$. Furthermore, we characterize when an involution would lift under the branched cover induced by an alternating action on $S_g$. Moreover, up to conjugacy, we derive conditions under which a given periodic mapping class is contained in a symmetric or an alternating subgroup of $\mathrm{Mod}(S_g)$. In particular, we show that symmetric or alternating subgroups can not contain irreducible mapping classes and hyperelliptic involutions. Finally, we classify the symmetric and alternating actions on $S_{10}$ and $S_{11}$ up to a certain equivalence we call weak conjugacy.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface of genus $g \geq 2$. | background 1.00 | 0.05 | 0.01 | asserted 0.73 |
| s1 | In this article, we derive necessary and sufficient conditions under which two torsion elements in $\mathrm{Mod}(S_g)$ will have conjugates that generate a finite symmetric or an alternating subgroup of $\mathrm{Mod}(S_g)$. | contribution 0.82 | 0.97 | 0.47 | asserted 1.00 |
| s2 | Furthermore, we characterize when an involution would lift under the branched cover induced by an alternating action on $S_g$. | result 0.63 | 0.95 | 0.08 | asserted 1.00 |
| s3 | Moreover, up to conjugacy, we derive conditions under which a given periodic mapping class is contained in a symmetric or an alternating subgroup of $\mathrm{Mod}(S_g)$. | result 0.59 | 0.97 | 0.16 | asserted 1.00 |
| s4 | In particular, we show that symmetric or alternating subgroups can not contain irreducible mapping classes and hyperelliptic involutions. | result 1.00 | 0.94 | 0.25 | asserted 1.00 |
| s5 | Finally, we classify the symmetric and alternating actions on $S_{10}$ and $S_{11}$ up to a certain equivalence we call weak conjugacy. | result 0.58 | 0.96 | 0.18 | asserted 1.00 |
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Original abstract
Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface of genus $g \geq 2$. In this article, we derive necessary and sufficient conditions under which two torsion elements in $\mathrm{Mod}(S_g)$ will have conjugates that generate a finite symmetric or an alternating subgroup of $\mathrm{Mod}(S_g)$. Furthermore, we characterize when an involution would lift under the branched cover induced by an alternating action on $S_g$. Moreover, up to conjugacy, we derive conditions under which a given periodic mapping class is contained in a symmetric or an alternating subgroup of $\mathrm{Mod}(S_g)$. In particular, we show that symmetric or alternating subgroups can not contain irreducible mapping classes and hyperelliptic involutions. Finally, we classify the symmetric and alternating actions on $S_{10}$ and $S_{11}$ up to a certain equivalence we call weak conjugacy.
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Since then, though the liftable mapping class groups under finite abelian covers have been widely studied [ref] , such an analysis is yet to be undertaken for covers induced by finite simple groups.
Generating the liftable mapping class groups of cyclic covers of spheres · Introduction · 0.86
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