Presentation complex

In group theory">group (mathematics)">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.

Properties

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.

Examples

Let be the two-dimensional integer lattice, with presentation
Then the presentation complex for G is a torus, obtained by gluing the opposite sides of a square, the 2-cell, which are labelled x and y. All four corners of the square are glued into a single vertex, the 0-cell of the presentation complex, while a pair consisting of a longtitudal and meridian circles on the torus, intersecting at the vertex, constitutes its 1-skeleton.
The associated Cayley complex is a regular tiling of the plane by unit squares. The 1-skeleton of this complex is a Cayley graph for.
Let be the Infinite [dihedral group], with presentation. The presentation complex for is, the wedge sum of projective planes. For each path, there is one 2-cell glued to each loop, which provides the standard cell structure for each projective plane. The Cayley complex is an infinite string of spheres.