Cohomology of arithmetic groups in GL(2) over definite quaternion algebras

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In this project, we will explore the cohomology of arithmetic groups arising from 2-by-2 matrices with entries coming from orders in definite quaternion algebras over the rational numbers. Such arithmetic groups give natural generalisations of the Bianchi groups and they act as isometries of hyperbolic 5-space. We will develop computer programs that will compute the cohomology of such arithmetic groups together with the action of Hecke operators. We will use these programs to explore various open conjectures that lie at the forefront of modern number theory.