Special Elliptic Isometries, Relative $$\mathrm{SU}(2,1)$$ SU ( 2 , 1 ) -Character Varieties, and Bendings
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Special Elliptic Isometries, Relative SU(2, 1)-Character Varieties, and Bendings Felipe A. Franco1
· Carlos H. Grossi2
Received: 20 February 2020 / Accepted: 3 September 2020 © Mathematica Josephina, Inc. 2020
Abstract We study relations between special elliptic isometries in the complex hyperbolic plane. Relations of lengths 2, 3, and 4 are fully classified. Some relative SU(2, 1)-character varieties of the quadruply punctured sphere are described and applied to the study of length 5 relations. Keywords Complex hyperbolic geometry · Character variety · Relative character variety Mathematics Subject Classification Primary 57S25; Secondary 51M10 · 30F60
1 Introduction Relations between automorphisms of a given geometric structure play an important role in the construction of manifolds/orbifolds endowed with that geometric structure. Consider, for instance, Poincaré’s Polyhedron Theorem, which is one of the few known tools for the construction of manifolds/orbifolds equipped with some model geometry (typically, a simply-connected Riemannian manifold). Roughly speaking, the theorem specifies conditions on a polyhedron with side-pairing isometries in the model space X such that the group H generated by these isometries is discrete and X /H is a manifold/orbifold M modelled on X . The group H is isomorphic to the fundamental
F. A. Franco: Supported by Grant 2014/00582-2, São Paulo Research Foundation (FAPESP), and by CNPq.
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Carlos H. Grossi [email protected] Felipe A. Franco [email protected]; [email protected]
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Departamento de Matemática, IMECC, Universidade Estadual de Campinas, Campinas, Brazil
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Departamento de Matemática, ICMC, Universidade de São Paulo, São Carlos, Brazil
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F. A. Franco, C. H. Grossi
group π1 (M) and the theorem provides an explicit presentation of H that comes from the combinatorial structure of the polyhedron with face-pairing isometries. This means that, in a certain sense, in order to construct a polyhedron with side-pairing isometries that have a chance of succeeding as a fundamental polyhedron, some relations between those isometries of X that will play the role of side-pairing isometries must be known a priori.1 More generally, the space of representations of the fundamental group π1 (M) in some group G of automorphisms of the model space modulo conjugation, i.e., the G-character variety of M, is closely related to the geometric structures on M inherited from the model space. Hence, it is natural to expect that (relative) character varieties are ubiquitous objects in geometry and that the many questions related to its structure (topology, Hitchin components, nature of the action of the mapping class group, etc.) are sources of great interest. They have been investigated by several authors, and an exhaustive list of references would be too long to compile; so, we only cite a few ones [1,7,9,12,14,15,19] which are closer to this paper. Here, our model space is the complex hyperbolic plane H2C with orientationpreserving isometries or, equivalently, the holom
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