Iwasawa Theory of Jacobians of Graphs

Gonet, Sophia R.

Algebraic Combinatorics, Tome 5 (2022) no. 5, pp. 827-848.

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 $\mu$ and $\lambda$ invariants by using the decomposition of a finitely generated module over the Iwasawa Algebra.

Mots-clés : Jacobian group, Iwasawa Theory, Laplacian, voltage graph, Picard group

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