The algebraic eigenvalue conjecture for discrete groups

Let $G$ be a discrete group. The algebraic eigenvalue property is the property that the eigenvalues under consideration for the combinatorial Laplacian are algebraic numbers. Algebraic eigenvalue conjecture. Every discrete group $G$ has the algebraic eigenvalue property. The paper establishes this property for several classes of groups, including the Linnell class, but the assertion for all discrete groups remains open.