A circulant matrix is a square matrix in which each row is obtained from its predecessor by a cyclic shift by one column. A cyclically presented group is one that is defined by a group presentation (ie a description of a group in terms of generators and defining relations) that admits a similar cyclic symmetry. Prominent examples are the Fibonacci groups, where the presentations' defining relations mimic the Fibonacci recurrence relation.
Questions that arise when studying cyclically presented groups ask when such a group is finite or perfect, when it is 3-manifold group or knot group; if so, what more can we say about the group. Knowledge of the determinant, rank, and the Smith normal form of a related circulant matrix can help to answer these questions.
In this talk I will discuss how well-known properties of circulant matrices have been used to gain insight into the structure of cyclically presented groups, and discuss recent and ongoing work with Vanni Noferini that is advancing the theory to provide new tools for studying such groups.