What is it about?

The study of symmetries is a fundamental part of Mathematics and Physics. Symmetries of surfaces in the sense of classical Euclidean or hyperbolic geometry are a classical field of study. Here we study symmetries of surfaces in the sense of projective geometry. We obtain a characterization of those surfaces which have more symmetries than the others.

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Why is it important?

Every finite group can be realizes as a group of symmetries of some surface. Surfaces for which this group of symmetries is very large have amazing algebraic, geometric, and number-theoretic properties. Here we make this precise in the context of projective geometry.


The moduli space of projective structures is much larger than the moduli space of hyperbolic structures. So I was surprised to find that surfaces with a maximal amount of symmetries are exactly the same as in the hyperbolic case. I would be interested in exploring the same question for other types of geometries.

Dr. Lorenzo Ruffoni
Tufts University

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This page is a summary of: Complex projective structures with maximal number of Möbius transformations, Mathematische Nachrichten, December 2018, Wiley,
DOI: 10.1002/mana.201700371.
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