Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In geometric group theory, a presentation complex is a 2-dimensional cell complex associated to any presentation of a group G. The complex has a single vertex, and one loop at the vertex for each generator of G. There is one 2-cell for each relation in the presentation, with the boundary of the 2-cell attached along the appropriate word. The fundamental group of the presentation complex is the group G itself. The universal cover of the presentation complex is a Cayley complex for G, whose 1-skeleton is the Cayley graph of G. Any presentation complex for G is the 2-skeleton of an Eilenberg–Maclane space K(G,1). Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces).

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