One variable equations over the lamplighter group

TL;DR

This paper proves that one-variable equations in the lamplighter group are decidable, provides an algorithm with super-exponential worst-case complexity, and shows that most equations can be decided in nearly quadratic time.

Contribution

It introduces a decidability result and an efficient algorithm for most cases of one-variable equations in the lamplighter group.

Findings

Decidability of one-variable equations in the lamplighter group

Super-exponential worst-case algorithm complexity

Nearly quadratic-time solution for most equations

Abstract

We prove that one variable equations in the lamplighter group $\MZ_2\wr \MZ$ are decidable and describe an algorithm for solving such equations. The algorithm has super-exponential time complexity in the worst case. We also show that, for most equations, decidability can be determined in nearly quadratic time; that is, the problem admits a nearly quadratic-time solution in the generic case.