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In classical mathematics, variables usually commute under multiplication. On the other hand, in this thesis we are interested in a setting in which variables do not always commute. Useful for representing this information are right-angled Artin groups (RAAGs).RAAGs are defined using graphs, i.e. sets of vertices and edges, where vertices represent variables and edges represent a commutative relationship between the connected vertices. RAAGs are often used as a tool to convert problems involving complex geometric phenomena into relatively simple algebra, as we can derive useful information directly from the geometric structure of the underlying graph. Bestvina-Brady groups (BBs) are normal subgroups of RAAGs, originally introduced by Bestvina and Brady to create subgroups of RAAGs which have exotic finiteness properties, such as subgroups which are finitely generated but not finitely presented.This thesis is focused on the problem of understanding splittings. Specifically, how to find an explicit description for some ways of decomposing the groups in terms of the geometry of the underlying graph. In this thesis, we review the findings by Groves and Hull for RAAGs and Chang for BBs, including details and full computations. The methodology used in this thesis relies upon three main topics: group theory, graph theory, and group actions on trees. In the last topic, we use Bass-Serre theory as a bridge between the group and graph theory.
Algebra, Group Theory, Graph Theory, Group actions on trees, Right-angled artin groups, bestvina-brady groups