Non-abelian simple groups act with almost all signatures

The study of Riemann surfaces and the groups which act on them is a classical area of research dating back to the latter half of the 19th century. Research in this field has wide-reaching implications in geometry and topology, algebra, combinatorics, analysis, and number theory through related topics such as the study of dessins d’enfants, mapping class groups, and graphs on surfaces. Today, this is still a rich area of research with many open questions. In this expository article we pose 78 open problems, contextualize them within the field, and discuss partial results or progress toward answering the questions, when relevant.

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Future directions in automorphisms of surfaces, graphs, and other related topics

Broughton

Paulhus

Wootton

2022

Contemporary Mathematics

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Planar representations of group actions on surfaces

Bozlee¹,

Lippert²,

Wootton

2022

Contemporary Mathematics

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The space of skeletal signatures was introduced as a simple but generally much coarser two-dimensional representation of the space of all signatures with which a group can act on a compact oriented surface. In this paper, we provide a complete characterization of the groups for which the skeletal signature space provides a complete and accurate picture of the structure of the full space of signatures. We show that such groups fall into three distinct families, and for two of these families, we show that for sufficiently large genus, the skeletal signature space depends essentially only on group order, and we explicitly describe these spaces. For the third family, we show that the skeletal signature space depends much more strongly upon group structure and provide some partial analysis of the structure of this space.

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Non-abelian simple groups act with almost all signatures

Cited by 2 publications

References 22 publications

Future directions in automorphisms of surfaces, graphs, and other related topics

Future directions in automorphisms of surfaces, graphs, and other related topics

Planar representations of group actions on surfaces

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