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. == References ==