Surfaces in Graphs of Groups and the Stable Commutator Length

In this thesis, we study stable commutator length (scl) in graphs of groups by analyzing surfaces in the corresponding graphs of spaces. We give a linear programming algorithm computing scl in a large class of graphs of groups including those with cyclic vertex and edge groups. The algorithm implies that the unit norm ball of scl is a rational polyhedron. We also establish a linear programming duality method to show sharp lower bounds of scl in graphs of groups, subject to a local n-relatively torsion-free condition on the inclusion of edge groups into vertex groups. As an application we prove a uniform sharp lower bound of scl in graph products.