Discrete vector fields and the cohomology of certain arithmetic and crystallographic groups
Bui, Anh Tuan
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This thesis has two main topics: Arithmetic groups and Crystallographic groups. The work on arithmetic groups SL(2, Z[1/m]) was motivated by a question of Kevin Hutchinson and also a generalization of the paper "On the cohomology of SL(2,Z[1/p])" by A. Adem and N. Naffah. Our main goal in relation to arithmetic groups is to present a new method for calculating the group homology and cohomology of the arithmetic groups SL(2, Z[1/m]). The latter topic, crystallographic groups, has been studied for many years and are still an active topic of research. Our main goals for crystallographic groups are to introduce: (i) new algorithms which attempt to calculate the group (co)homology of certain crystallographic groups; (ii) a method for computing the cohomology ring structures for those groups where the algorithms of (i) are successful.