Iwasawa Theory of Jacobians of Graphs

Abstract

The Jacobian group (also known as the critical group or sandpile group) is an important invariant of a finite, connected 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). A specific type of covering graph, called a derived graph , that is constructed from a voltage graph with voltage group G is the object of interest in this paper. Towers of derived graphs are studied by using aspects of classical Iwasawa Theory (from number theory). Formulas for the orders of the Sylow p -subgroups of Jacobians in an infinite voltage p -tower, for any prime p , are obtained in terms of classical µ and λ invariants by using the decomposition of a finitely generated module over the Iwasawa Algebra.