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.

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. 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.

Text derived from https://en.wikipedia.org/w/index.php?oldid=971408180 under CC BY-SA 4.0; modifications have been made.