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Homotopy theory.

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In the first part of this thesis we study the cohomology of split extensions of groups and Lie algebras by investigating the spectral sequence associated to a split extension of Hopf algebras. We develop methods to analyse the differentials of this spectral sequence, and use these methods to show that the spectral sequence collapses at the second page in certain instances. The second part of this thesis revolves around finiteness properties of classifying spaces for families of subgroups. We show there exist finite dimensional classifying spaces with virtually cyclic stabilizers for countable elementary amenable groups of finite Hirsch length and certain groups that act by discrete orbits and semi-simple isometries on finite dimensional CAT(0)-spaces. We also investigate the notion of geometric dimension of groups for the family of finite subgroups and the family of virtually cyclic subgroups. In particular, we study the behavior of these invariants under group extensions, their relationship and their connection with various notions of cohomological dimension involving modules over orbit categories and Mackey functors.