Online

Gonet, Sophia Rose

Mathematics and Statistics

2021

Ph. D.

The Jacobian group (also known as the critical group or sandpile group) is an important invariant of a graph X; it is a finite abelian group whose cardinality is equal to the number of spanning trees of X (Kirchhoff's Matrix Tree Theorem). This dissertation proves results about the Jacobians of certain families of covering graphs, Y, of a base graph X, that are constructed from an assignment of elements from a group G to the edges of X (G is called the voltage group and Y is called the derived graph). The principal aim is to relate the Jacobian of Y to that of X.We develop the basic theory of derived graphs, including computational methods for determining their Jacobians in terms of X. Of particular interest is when the voltage assignment is given by mapping a generator of the cyclic group of order d to a single edge of X (all other edges are assigned the identity), called a single voltage assignment. We show that, in general, the voltage group G acts as graph automorphisms of the derived graph Y, that the group of divisors of Y becomes a module over the group ring Z[G], and that the Laplacian endomorphism on the group of divisors of Y---which is used to compute the Jacobian of Y---can be described by a matrix with entries from Z[G] called the voltage Laplacian. Using this and matrix computations, we determine both the order and abelian group structure of the Jacobian of single voltage assignment derived graphs when the base graph X is the complete graph on n vertices, for every n and d. When G is abelian, the determinant of the voltage Laplacian matrix is called the reduced Stickelberger element; and it is shown to be a power of two times the graph Stickelberger element defined in the literature in terms of Ihara zeta-functions. Also using zeta-functions, we develop some general product formulas that relate the order of the Jacobian of Y to that of X; these formulas, that involve the reduced Stickelberger element, become very simple and explicit in the special case of single voltage covers of X. We adapt aspects of classical Iwasawa Theory (from number theory) to the study of towers of derived graphs. We obtain formulas for the orders of the Sylow p-subgroups of Jacobians in an infinite voltage p-tower, for any prime p,in terms of classical μ and λ invariants by using the decomposition of a finitely generated module over the Iwasawa Algebra.