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.

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 $$K(G,1)$$.

Examples
Let $$G= \Z^2$$ be the two-dimensional integer lattice, with presentation


 * $$ G=\langle x,y|xyx^{-1}y^{-1}\rangle.$$

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 $$\Z^2$$.

Let $$G = \Z_2 *\Z_2$$ be the Infinite dihedral group, with presentation $$\langle a,b \mid a^2,b^2 \rangle$$. The presentation complex for $$G$$ is $$\mathbb{RP}^2 \vee \mathbb{RP}^2$$, 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.