Smith forms of matrices in Companion Rings, with group theoretic and topological applications

Abstract

Let R be a commutative ring and $g\left(t\right)\in R\left[t\right]$ a monic polynomial. The commutative ring of polynomials $f\left(C_g\right)$ in the companion matrix $C_g$ of $g\left(t\right)$, where $f\left(t\right)\in R\left[t\right]$, is called the Companion Ring of $g\left(t\right)$. Special instances include the rings of circulant matrices, skew-circulant matrices, pseudo-circulant matrices, or lower triangular Toeplitz matrices. When R is an Elementary Divisor Domain, we develop new tools for computing the Smith forms of matrices in Companion Rings. In particular, we obtain a formula for the second last non-zero determinantal divisor, we provide an $f\left(C_g\right)\leftrightarrow g\left(C_f\right)$ swap theorem, and a composition theorem. When R is a principal ideal domain we also obtain a formula for the number of non-unit invariant factors. By applying these to families of circulant matrices that arise as relation matrices of cyclically presented groups, in many cases we compute the groups' abelianizations. When the group is the fundamental group of a three dimensional manifold, this provides the homology of the manifold. In other cases we obtain lower bounds for the rank of the abelianization and record consequences for finiteness or solvability of the group, or for the Heegaard genus of a corresponding manifold.