Derived Categories of Surfaces and Group Actions.

This thesis focuses on two distinct projects on the bounded derived category of coherent sheaves of surfaces and group actions from different directions. The first project studies bielliptic surfaces, which arise as quotients of products of elliptic curves by a finite group acting freely. We prove a structure theorem describing the group of exact autoequivalences of the bounded derived category of coherent sheaves on a bielliptic surface over C. We also list the generators of the group in some cases. The second project studies semi-orthogonal decompositions of the bounded equivariant derived category of a surface S with an effective action of a finite abelian group G. These semi-orthogonal decompositions are constructed by studying the geometry of the quotient stack [S / G]. We produce new examples of semi-orthogonal decompositions of the equivariant derived category of surfaces with a finite abelian group action. We give a new proof of the Derived McKay correspondence in dimension 2. Using this, we construct semi-orthogonal decompositions of the equivariant derived category of C^2 with an effective action of the Dihedral group D_2n. Moreover, we show that these semi-orthogonal decompositions satisfy a conjecture of Polishchuk and Van den Bergh.