Computability Models: Algebraic, Topological and Geometric Algorithms

TL;DR

This paper investigates the decidability of the discreteness problem for certain subgroups of $PSL(2,ullet)$, showing that the answer varies with the computational model used, bridging computability theory and group theory.

Contribution

It demonstrates that the decidability of the discreteness problem depends on the computational model, integrating computability and group theory insights.

Findings

Decidability varies with the computational model.

Provides background connecting computability and group theory.

Highlights open problems in subgroup discreteness.

Abstract

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