We study the problem of density of polynomials in the de Branges spaces ℋ($E$) of entire functions and obtain conditions (in terms of the distribution of the zeros of the generating function$E$) ensuring that the polynomials belong to the space ℋ($E$) or are dense in this space. We discuss the relation of these results with the recent paper of V. P. Havin and J. Mashreghi on majorants for the shift-coinvariant subspaces. Also, it is shown that the density of polynomials implies the hypercyclicity of translation operators in ℋ($E$).

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We give new examples of hyperbolic and relatively hyperbolic groups of cohomological dimension d for all d ≥ 4 (see Theorem 2.13). These examples result from applying CAT(0)/CAT(−1) filling constructions (based on singular doubly warped products) to finite volume hyperbolic manifolds with toral cusps. The groups obtained have a number of interesting properties, which are established by analyzing their boundaries at infinity by a kind of Morse-theoretic technique, related to but distinct from ordinary and combinatorial Morse theory (see Section 5).

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It is known that finitely generated FI-modules over a field of characteristic 0 are Noetherian. We generalize this result to the abstract setting of an infinite EI category satisfying certain combinatorial conditions.