Equations in Wreath Products of Abelian Groups

Abstract: Solving equations is among the most classical and fundamental questions in Mathematics. Typically, equations are solved over fields of numbers such as the rational, real or complex numbers. Solving equations over the ring of integers (Hilbert’s 10th problem) is famously known to be undecidable. If, instead of considering the two operations of addition and multiplication of a ring, we restrict ourselves to a single one, we very naturally arrive at the Diophantine problem over a group: given a system of/a single equation(s) containing variables and constants (from the group), can we decide whether we may substitute the variables with some group elements such that the equation holds? In many groups this problem is generally undecidable. However, we can restrict the question further and consider only quadratic equations (where every variable appears at most twice, counting positive and negative instances). In the talk, we will discuss (ongoing) joint work with Ruiwen Dong and Leon Pernak to solve quadratic equations in (restricted) wreath products of abelian groups. The talk will contain a short introduction on equations over groups, the lamplighter group and its generalization to wreath products of abelian groups. Then we will look at the case of nonorientable equations over these groups.

Mathematics

Audience: researchers in the topic

ANTLR seminar

Series comments: This is the Algebra, Number Theory, Logic and Representation theory seminar.