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The discreteness problem for finitely generated subgroups of$PSL(2,\mathbb{R})$ and $PSL(2,\mathbb{C})$ is a long-standing open problem. Inthis paper we consider whether or not this problem is decidable by analgorithm. Our main result is that the answer depends upon what model ofcomputation is chosen. Since our discussion involves the disparate topics ofcomputability theory and group theory, we include substantial backgroundmaterial.

Computability Models: Algebraic, Topological and Geometric Algorithms