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).

Example

Let "G" =Z² be the two-dimensional integer lattice, with a presentation

: $G=langle\; x,y|xyx^\{-1\}y^\{-1\}\; angle.$

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 Z².

References

* Roger C. Lyndon and Paul E. Schupp, "Combinatorial group theory". Reprint of the 1977 edition (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89). Classics in Mathematics. Springer-Verlag, Berlin, 2001 ISBN 3-540-41158-5