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The authors investigate the correspondence between holomorphic automorphic forms on the upper half-plane with complex weight and parabolic cocycles. For integral weights at least 2 this correspondence is given by the Eichler integral. The authors use Knopp's generalization of this integral to real weights, and apply it to complex weights that are not an integer at least 2. They show that for these weights the generalized Eichler integral gives an injection into the first cohomology group with values in a module of holomorphic functions, and characterize the image. The authors impose no condition on the growth of the automorphic forms at the cusps. Their result concerns arbitrary cofinite discrete groups with cusps, and covers exponentially growing automorphic forms, like those studied by Borcherds, and like those in the theory of mock automorphic forms.

Automorphic forms. --- Holomorphic functions. --- Cohomology operations.

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These proceedings comprise two workshops celebrating the accomplishments of David J. Benson on the occasion of his sixtieth birthday. The papers presented at the meetings were representative of the many mathematical subjects he has worked on, with an emphasis on group prepresentations and cohomology. The first workshop was titled "Groups, Representations, and Cohomology" and held from June 22 to June 27, 2015 at Sabhal Mòr Ostaig on the Isle of Skye, Scotland. The second was a combination of a summer school and workshop on the subject of "Geometric Methods in the Representation Theory of Finite Groups" and took place at the Pacific Institute for the Mathematical Sciences at the University of British Columbia in Vancouver from July 27 to August 5, 2016. The contents of the volume include a composite of both summer school material and workshop-derived survey articles on geometric and topological aspects of the representation theory of finite groups. The mission of the annually sponsored Summer Schools is to train and draw new students, and help Ph.D students transition to independent research. .

Representations of groups --- Cohomology operations --- Operations (Algebraic topology) --- Algebraic topology --- Algebra. --- Topological Groups. --- Group theory. --- Category Theory, Homological Algebra. --- Topological Groups, Lie Groups. --- Associative Rings and Algebras. --- Group Theory and Generalizations. --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Groups, Topological --- Continuous groups --- Mathematics --- Mathematical analysis --- Category theory (Mathematics). --- Homological algebra. --- Topological groups. --- Lie groups. --- Associative rings. --- Rings (Algebra). --- Algebraic rings --- Ring theory --- Algebraic fields --- Rings (Algebra) --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Homological algebra --- Algebra, Abstract --- Homology theory --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Topology --- Functor theory