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- W3157369426 abstract "We study a particular case of maximal homomorphisms from a surface group into a Hermitian Lie group of tube type, which we call integral maximal.In the first part, we deal with the case when the Lie group is locally isomorphic to the group of isometries of the hyperbolic plane. In this case, integral maximal homomorphisms induce hyperbolizations of the initial surface and we relate them to spin structures on Riemann surfaces, that is to line bundles whose tensor power is isomorphic to the tensor product of the canonical bundle and a given divisor. Fixing such an integral maximal representation, we associate to each geodesic an integer modulo a fixed integer, its translation number. We then give, when the surface is closed, the asymptotic growth of the number of geodesics with given translation number.In the second part, we study the general case of an arbitrary Hermitian Lie group of tube type. Fixing a specific finite cover of such a Lie group, we call the representations into the cover spin representations and we show that the space of integral maximal spin representations is homeomorphic to the product of the space of maximal representations into the initial Lie group and an explicit subspace of homomorphisms from the first homology group with integer coefficient of the unit tangent bundle of the surface into a finite cyclic group.The homeomorphism we construct is moreover mapping class group equivariant so that we naturally study the action of the mapping class group on the space of homomorphisms from the first homology group of the unit tangent bundle of the surface into a finite cyclic group.Finally we apply these results to count the number of connected components of diagonal representations into some Lie groups locally isomorphic to the symplectic group." @default.
- W3157369426 created "2021-05-10" @default.
- W3157369426 creator A5058062535 @default.
- W3157369426 date "2019-06-25" @default.
- W3157369426 modified "2023-09-23" @default.
- W3157369426 title "Spin representations for Hermitian Lie groups" @default.
- W3157369426 hasPublicationYear "2019" @default.
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