# 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.

## Example

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

## References

• R. Brown and J. Huebschmann, Identities among relations, in Low dimensional topology, London Math. Soc. Lecture Note Series 48 (ed. R. Brown and T.L. Thickstun, Cambridge University Press, 1982), pp. 153-202.
• Hog-Angeloni, C., Metzler, W. and Sieradski, A.~J. (eds.). Two-dimensional homotopy and combinatorial group theory, London Mathematical Society Lecture Note Series, Volume 197. Cambridge University Press, Cambridge (1993).