Presentation complex

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.
Examples.
Let formula_2 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 formula_4. 
Let formula_5 be the Infinite dihedral group, with presentation formula_6. The presentation complex for formula_7 is formula_8, 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.